In what cases can we begin noun clauses with a preposition?

Your question is much broader than the examples you give. I assume what you mean is a noun phrase. There is no such thing as a noun clause. You might care edit your question accordingly. This answer is directed to what I take to be what you are asking.

There used to be a rule that forbade what was called a 'dependent preposition'. So it was when I was at school over 50 years ago. According to this rule it was wrong to leave a preposition to the end of a sentence or clause. There was some reason for this. So the following sentence, in which a child complains to a parent for bringing to the bedtime reading a book s/he did not like sound ridiculous as well as almost unintelligible.

What did you bring that book I didn't want to be read to out of up for?

However, if I were to obey the prohibition of 'dependent prepositions', something equally preposterous results.

For what did you bring up that book out of which I did not to be read?

So both extremes have severe potential problems. So what are we to do?

If we go back to basics, the problem starts with the word itself: 'preposition. The Latin derivation indicates the placement (position) of a word before. So Greenbaum in the Oxford English Grammar [OEG] (4.31) says:-

Typically, prepositions function as the first constituent of a prepositional phrase.

But note the word typically: not necessarily! He then goes on to provide a useful list of English prepositions and additional list of the most commonly used.

He goes on the explain [OEG 4.43] that With wh-pronouns and determiners (who, whom, whose, which, what)

The wh-words or phrases with the wh-word are initial in their their sentence or clause. Hence, if they are not the subject, they are generally fronted .

He gives an example: A) a simple sentence without a wh-word, then B) one with a wh-word, in this case an exclamation.

A) They gave me an expensive gift for my birthday.

B) What an expensive gift they gave me for my birthday.

So we have two principles that speakers of English tend to follow: they tend to put a preposition before the noun it modifiers; and they tend to 'front a wh-word. Simple. So what if I want the preposition to modify a wh-word? Lets take your first example. First I shall take a non-wh-word version.

1. You need to agree with vegetarianism and not with veganism.

Now to turn it into your question, following the 'rule': wh-word to be fronted, and preposition to come before its noun. Well, we are asking about the thing we agree with and the thing we do not agree with. Oh dear. There is a problem. The preposition 'with' points on two directions at once. It also goes with the verb agree, almost like a suffix. Which way do I go: move 'with' up front to modify its noun, or leave it with it partner verb, agree?.

1a. You need to choose with what you agree and with what not.

1b. You need to choose what you agree with and what ....

We have a problem either way. in 1a, the ellipse at the end does not work. It has to be "...and with what you don't ." This solution will just about work for 1b: "You need to choose what you agree with and what you don't." But to me it feels incomplete. So probably we should end up with 1c.

1c. You need to choose what you do and what you don't agree with.

You could just as well go back to 1d.

1d. You need to choose with what you do and don't agree.

Geriatrics like me might be drawn to 1d, but younger people would append the preposition to the verb.

Your version of 2. is correct, though a little old-fashioned, as is 3. Except for the final "and when" in 2. That is because it tries to pull the reader in two directions at once: "when ...what? I am asking. "If I am choosing not to work with someone, what has 'when not to work with them' got to do with anything?'. Of course we can figure out that it the writer or speaker intends the 'and when' to apply only to those whom I choose to work with or with whom I choose to work. But the sentence is clumsy. It jars on the logical nerves. You could clarify by saying "... and when to work with them". But that only makes matters worse. That clause has to 'jump over' its closer grammatical neighbour to the clause on the other side.

As to whether we use 'with whom to work' or 'whom to work with', that in part becomes a matter of generation and even taste. Also, a typical British move starts. If the meaning is clear, why bother with the accusative at all. Why not 'who to work with'. For al long time, people like me ground their teeth and complained of lack of grammar in the younger generation. But over time the use of 'whom' and 'whose' has been in decline.

4. is problematic for a difference. Unlike the preposition 'with', which has become 'fused' to verbs like 'agree', the preposition 'until' is not becoming fused with other verbs. There is nothing strictly wrong with the sentence. You could not delay the 'until' and write "Does anyone know when that shop works until?'. Even the original sentence, though grammatically sound, feels awkward. I should probably was to rephrase it completely: perhaps as "Does anybody know when that shop shuts?"

Much the same can be said about 5 and 6. They are correct, but old fashioned. Most people would prefer ".. who built it..".

Neither 7 nor 8 nor 9 poses any problem grammatically. Many might prefer to put them differently (less formally) but that is a matter of style and preference.

Sentence 10 illustrates a point already made. The preposition for has become attached to the end of the verb go so that the two make a virtual compound verb. So most people today would write or say

10. I can't make my mind up which to go for, the red or the blue.

There is one limiting rule about the separation of the preposition from its allotted wh-word. Do not separate it’s partner by too much.

11. What criteria can we judge the relative merits of electing a president according to popular vote versus the electoral college by?

This is intolerable. It is the paradigm of the dreaded ‘dangling preposition’.