“I understand the logic of why G-d can not be plural and does not change, except for how the Bible speaks of Him appearing to people.  If G-d is truly incorporeal [without physical form nor subject to time/space] why does the Bible speak of Him coming to certain prophets in the form of a man?”     See further:  Biblical Words and Idioms Misunderstood,  Singularity of G-d, To See G-d 

Very important question.  A critical principle to always remember is that the Torah speaks in the language of men.  This is a teaching of the Sages of Israel, men of the Moses’ Court.  Because the Almighty is absolutely unlike anything in creation, it is impossible for us to refer to Him in any way at all unless we make use of metaphors, homonyms, and idioms based upon the things we commonly see and experience in the realm of creation that we live in.  So, on the one hand, while it is true that nothing is comparable the L-RD [Isaiah 40:18; Isaiah 40:25; Isaiah 46:5; Jeremiah 10:6-7; Deuteronomy 33:26; Exodus 8:10; Psalm 40:5], we must make comparisons to things in creation in order to ever refer to Him.  HOWEVER, heaven forbid that we should forget for a moment that in reality no comparison is truly applicable to Him. We are limited to human speech which is itself subject to time and space.  No metaphor and no vision ever touch a “speck” of His Unimaginable Exaltedness.  Notice, I am not even able to convey a thought concerning His Incorporeity without using corporal terms of physical things as metaphors: I said “speck.” 

Concerning appearances of G-d to prophets and others in the Bible, keep three things in mind:   

1) The term Elohim does not mean divinity like the word ‘G-d’ in English.  The word Elohim implies power and authority.  This word is sometimes used in reference to angels, idols, judges, and powerful men.  Sometimes the Bible may use the term Elohim without intending to refer to the Creator at all.  In the same place the translators of Bible version you’re using may have chosen to translate it to English as ‘G-d’ for whatever reason.  Sometimes even special names of G-d are used in the personal names or titles of human beings as a way of honoring the Almighty.  Check to see if the passage where you think G-d has become physical is possibly referring to someone else with the term Elohim.

2) Some places in the Bible it may seem that the Almighty appeared to someone as a human being, but then in another place in the Bible it refers to the same incident as having been a vision.  A vision is not physical reality. Visions occur in the mind of the prophet but appear as if they are physical reality.  If a prophet has a vision of G-d as a man and you want to take it literally, then why don’t you take the other metaphorical things in the vision literally as well?  Choose the most logical and consistent method of understanding the Bible.

Hosea 12:10 “And I have spoken unto the prophets, and I have multiplied visions, and by the hand of the prophets I use similes.”

3) This section is related to section one.  A created thing can act as a representative of the Creator, whether the created thing is a human being, an angel, a burning bush, or something else.  In such a case the created thing may speak with the authority of the Almighty while acting as His messenger.  Moses is even called Elohim at one point, and Aaron is said to be his prophet [Exodus 7:1; The word 'like' or 'as' is added in the translations but not in the Hebrew].

The pillar of fire and the pillar of cloud that lead the Israelites through the desert are at times referred to as ‘the L-RD [Number 14:14; Deuteronomy 5:4].’  At other times the pillar of cloud and the pillar of fire are referred to as a mal’akh [a messenger, aka ‘angel [Exodus 14:19].’]  If the pillar of cloud or of fire is actually G-d, then how could G-d have said to the Israelites in Deuteronomy 4:15 that they didn’t see any form of G-d in the desert when He spoke to them from the fire? Clearly He is teaching them that even though the fire has been referred to as the L-RD, nonetheless the fire is NOT a form of G-d.  The use of the phrases "face to face, eye to eye, etc.." in the verses are all only metaphorical ways of expressing how overwhelmingly blatant the truth of the L-RD's Existence was perceived and how clearly the L-RD's Instruction [Torah] was revealed at that specific time. As stated, these manifestations are also called mal'akh [messenger/angel]. How could they be called mal’akh [messenger] if they are G-d? In fact, Isaiah 4:5 flat out states in plane Hebrew that the fire and the cloud are CREATED.   Besides, a messenger does not send itself. If a messenger sends itself then it is no longer merely the conveyer of the messenger. It would then be the originator of the message.  If the pillar of fire and the pillar of cloud were literally physically G-d, then there wouldn’t be indications to the contrary.  If one wants to say that any human or angelic creature who speak as messengers of G-d were literally physically G-d, then why should they say anything differently concerning the burning bush or any other revelation of G-d?  But the Israelites did not go from the creation worshipping idolatry of the Egyptians to an idolatry of worshipping a cloud-god, a fire-god, a burning-bush-god, or another man-god. Don't be ignorant of the fact that among the various idolatrous beliefs of Egypt was that their Pharaoh King was both man and deity united in one human form. Also, among their cattle, one of their most beloved objects of worship was the lamb. This is why the Egyptians hated shepherds, why the Hebrews had to make sacrifices distant from the Egyptians, and why the Hebrews had to slay a lamb to be saved. By slaughtering the lamb the Hebrews merited deliverance from slavery. Why? Their faith in the Unity of the Almighty and rejection of idolatry reached such a level that it was able to publicly manifest itself in the killing of the Passover lamb, a historical symbol of Egyptian idolatry, Exodus 8:25-27. Christians unknowingly resurrect the idea of both the god-man king and the lamb-god of the Egyptians and equate it with the L-RD of Israel. This is a horrible recurrence of the sin of the golden calf, Exodus 32:4.  The Temple and Tabernacle are called the “House of G-d,” and the Almighty says He “dwells” there [1 Chronicles 17:3-5, Deuteronomy 12:11], but then in other places the Bible clearly states that the Temple does not contain Him, nor can the highest of heavens contain Him [2 Chronicles 2:6, 6:18; 1 Kings 8:27; Isaiah 66:1].  If there is obviously there is a deeper meaning to His “dwelling” in the Temple other than the literal interpretation, along with the other Bible passages that teach against a literal 'dwelling' of the Creator in any form within time/space, then doesn't it make sense to say that the same is likely true with regard to the Creator 'dwelling in' the pillar of cloud or fire, in the people of Israel, or in any man or any other thing. The basic understanding of this 'DWELLING' is that it simply refers to a lasting expression/manifestation of His providence and holiness over the particular place, person, nation, or thing.  The Israelites understood that the Creator reveals or manifests Himself with created revelations of Himself.  They did not confuse a revelation with the One who was revealing.  The Torah, the Law of Moses, is also a revelation of the Almighty.  Is the Torah G-d in book form?  What absurdity.  Rather, just as an artist’s painting reveals expresses the artist and yet the two are separate things, so also the Almighty is not the same as his revelations and expressions in creation.  Just as one can read a book written by John Doe about John Doe and then point to the book and metaphorically say, “That’s John Doe,” so also at times the manifestation of the Creator may be called by His Name.  In such a case it is understood that the Name is not referring to the manifestation itself, but rather to the ONE who created the manifestation, to the ONE who sent the messenger, to the ONE who revealed the revelation.  The holy men of the Bible both were informed of things via metaphor and allegory, and also taught things by way of metaphor and allegory.  Let’s not confuse the book for the Author.  Let’s not praise the painting instead of the Painter.

The issue of “divine manifestations” in the Bible is one of the main “proofs” Christianity relies upon to say that Jes’us is G-d as a man.  Christians use the revelations of G-d in the Only Testament [‘old testament’] similar to the way they use prophecies as “proof” that Jes’us is Messiah.  First of all, concerning the way they use references to ‘divine manifestations’ and prophecies in the Tanakh [O.T.], even IF we were to admit that all the prophecies they refer to really are messianic prophecies, and all the ‘manifestations’ they refer to truly represent occurrences when the Almighty really dwelt on earth with man, EVEN STILL they have proved nothing of Christianity.  Why?  Because even admitting all this, where in the texts of all these ‘divine manifestations’ and prophecies does it say that these prophecies and manifestations are of Jes’us?  And where in any of these prophecies or manifestations does it say that a person can have eternal life ONLY if he believes in Jes’us?  Such references to Jes’us and Christian doctrine of salvation are in none of these prophecies or manifestations.  Take Isaiah 53 for example.  Assuming this is a prophecy about the Messiah making atonement, where does it say that Jes’us is this same Messiah?  Where in Isaiah 53 does it say that those who are unaware of the Servant do not have eternal life?  It’s not written.  One should realize that there have been other religious groups who also took these manifestations as literal appearances of G-d as a man, and other groups who also claimed some of the same prophecies Christianity claims as fulfillment, who do not recognize Jes’us and do not accept Christianity.  Islam claims certain of the prophecies Christians believe refer to Jes'us as references to Muhammad.  Certain heretical Jewish groups have taken certain revelations of G-d literally just like Christians do, without for a moment seeing an implication that Jes’us is Messiah in their literal interpretation of such revelations.  The best way to deal with such “proofs” as these is not to try and answer every one.  If we must explain each “proof” before a person will realize that the manifestations are not the same as G-d Himself, then even if he accepts that 99 of 100 of the revelations are not actually G-d, just that last remaining revelation that he doesn’t understand will hold him back from realizing the absolute incorporeity of the Almighty.  Instead, the better method to use is to explain to principles of reasoning.  Consistency.  I understand that the Almighty is absolutely not plural and that His existence infinitely transcends creation - time and space.   While realizing His Unity and incorporeity I can still understand that the Almighty expresses and reveals His existence to creation in various ways.  Along with understanding that the Creator is eternally non-physical, I can still provide a meaningful and consistent understanding of the ‘appearances’ of G-d to different prophets.  The question is:  Can those who believe the appearances G-d to the prophets were LITERAL accounts of the Creator becoming a man [or fire, or cloud, or smoke..etc..] make this understanding of their's consistent with the many other verses in the Bible which teach the opposite?  I have yet to see from them any rational explanations which would make their belief in G-d being physical consistent with the other verses [and basic common sense] that indicate that the Creator is not within time and space, and that He is unchanging.  Generally people who hold such beliefs either “answer” by giving more verses as examples of their belief without explaining the logical contradiction of their ideas, change the subject, or try to walk away.  Only the few who are humble enough to face the logical contradictions and inconsistencies in such beliefs of G-d becoming a man in pure honest pursuit of Truth come to the ego-shrinking realization of the Infinite Awesomeness of His Exalted Majesty.  May it be that the few become many.  "A person who pursues righteousness and loving-kindness will find life, righteousness, and honor." - Proverbs 21:21  “..you will find Him if you seek for Him with all your heart and all your being.” - Deuteronomy 4:29

See further:  Salvation in the Torah

People were and are always atoned for, and go to 'heaven,' in the same way.  The Almighty does not change; Neither does the essential way to 'come before Him.'  The way this 'coming before Him' is expressed has at times changed, depending on particular conditions.  NEVER confuse the manner in which a thing is expressed with that which is being expressed itself.  There is a difference between a revelation and what is being revealed.  I'll begin to answer your question with a question:  How were people atoned for after the destruction of the 1st Temple but before the rebuilding of the 2nd Temple, at which time there were no sacrifices?  How were the people of Nineveh saved in the book of Jonah?  They turned their hearts back toward their Maker and humbled themselves before Him, calling on Him with sincerity.   "The L-RD is near to all who call upon Him, to all who call upon Him with sincerity. He will fulfill the desire of those who revere Him; He will hear their cry and save them as well." - Psalm 145:18-19     "You caused judgment to be heard from heaven; The earth was afraid and was quiet; When the All-powerful Authority arose to judgment, to save all the humble of the earth. Selah." - Psalm 76:8-9  We believe sacrifices are an expression or revelation of an atonement which already occurred when the person repented.  Since the sacrifice is such a powerful expression of the slaying of our rebellious nature, and the ultimate result of rebellion had we not repented - death, therefore the sacrifice is symbolically identified with atonement, similar to the way many Christians consider baptism to be symbolically identified with 'dieing to one's sins.'

"The other was about the judgments of the Sanhedrin. The One who is, instituted a body of Judges for His people and I believe that as long as they are acting under His authority their judgments are true. Of course, their judgment must not contradict Scripture. Today, a judge does not make a law but interprets one that is already made. It seems that the Sanhedrin sometimes made laws. Are what they said equal to Scripture?"                  See further:  What is Oral Law?;      Misconceptions about Jewish Rituals;

In short, the answer to your question is 'NO.'  None of the rulings of the Sanhedrin are equal to the Law of Moses.  However, according to the Law of Moses we are obligated to keep their rulings.  In fact, the Almighty considers their righteous judgments as His own - Deuteronomy 1:17.  We have the written Instruction (Torah sh'b'khtav) and oral Instruction (Torah sh'b'al'peh).  Neither one of these two can be applied without the other:

The written Instruction [written 'Law'] most specifically refers to the first 5 books of the Bible, also called the 'Law of Moses.'  The Almighty dictated these to Moses while we were in the Sinai wilderness, with great wonders and amazements that the whole people of Israel witnessed.  These 5 books together form the Constitution of the people of Israel.

The oral Instruction [oral 'Law'] refers firstly to the authoritative meaning of the 5 books of Moses.  The meaning of the words and references recorded in the 5 books of Moses, which may not necessarily be clear in the written text alone, were divinely revealed to Moses along with the written Instruction.  Moses orally taught the meanings of the words, phrases, and references in the written Torah to the members of the court he was commanded to establish.  This is halakhah l'Mosha me'Seynai [halakhah to Moses from Sinai].  Included in this oral Instruction given to Moses were principles by which the judges of the court he established would be able to arrive at authoritative applications of the Law of Moses in future times when new situations would arise.  The rulings of this court are called halakhah d'Rabanan [halakhah of the Rabbis].  There are still finer distinctions to be made, but that is a broader discussion.  Essentially, we have the CORE - the Law of Moses, then we have the authoritative rulings of the Sanhedrin on how to carry out / live out the Law of Moses.  Although the rulings of the Sanhedrin are not considered equal to the Law of Moses, at the same time it is an extension of the Law of Moses.  The Law of Moses is the root/trunk of the tree.  The rulings of the Sanhedrin are the branches and leaves.  While the Law of Moses will not change, nor be added to or taken away from - the rulings of the Sanhedrin can be pruned and trimmed like branches.  New rulings can be made and old rulings can be nullified - though ONLY with the authority of the Sanhedrin.  The essential difference between the 613 commands of the Law of Moses and the 7 commands and many other different types of rulings of the Sanhedrin is that the 613 commands are eternally binding in so far as we can carry them out, and they were directly and divinely given to Moses by the Almighty; Whereas, all the rulings of the Sanhedrin are not intrinsically everlasting.  They only exist as long as the Sanhedrin sees the need for them.  Each of the rulings of the Sanhedrin only exists for the sake of carrying out and applying the 613 commands of the Law of Moses to our daily lives.  The rulings of the Sanhedrin are not directly revealed to creation by the Almighty in the manner that the 613 of the Law of Moses were revealed to Moses.  The rulings of the Sanhedrin are binding and obligatory only because they are derived by the  Sanhedrin using divinely ordained principles and because the Sanhedrin has been give divine judicial authority by the Law of Moses.  "You shall not show favoritism in judgment; you shall hear the small & great alike; you shall not fear man's presence, for the judgment is G-d's.." <- Deuteronomy 1:17; 17:11 -> "According to the dictation of the law (oral law) which they teach you, according to the judgment which they tell you, you shall do; you shall not turn aside to the right or to the left from the verdict which they proclaim upon you."   Exodus 18:15-26;   Deuteronomy 1:13-18, 17:8-13;  "The Torah of the L-RD is whole, restoring the soul; the testimony of the L-RD is trustworthy, making the simple wise."  - Psalm 19:7

Could you live according to the Constitution of the United States without establishing and obeying the Judicial System and courts of law?  It's impossible to do so, for the Constitution speaks of establishing such a system.  Imagine following the Constitution' while ignoring the Judicial System.  This would actually be rebelling against the Constitution, not keeping it.  The Torah is our Constitution, and the Sanhedrin with its' subsisting courts is our judicial system.  May it be reestablished in our days with the coming of Messiah.  Then the sovereignty of the ONE All-powerful Creator will again be recognized in all the Earth.