Is [pronoun + (some verbs) + all] generally preferred?
I think it might depend on the pronoun and speaker. I say "Y'all", so "you all (auxiliary verb)" sounds more natural than "you (auxiliary verb) all", though both are acceptable.
However, "we all (auxiliary verb)" and "they all (auxiliary verb)" sound odd to me. (Maybe because they sound too similar to we'll or they'll?) In those cases, I'd strongly prefer "we (auxiliary verb) all" or "they (auxiliary verb) all". I think another reason for the difference is that we and (usually) they is unambiguously plural, while "you" could be singular.
Like you said, "You have to all go to school" sounds the least natural, but it's acceptable.
"And we have to all shoulder the responsibility..." - Obama, 2013
Which other copular verbs should be used in this structure?
I'm unable to think of any. I suspect the source meant auxiliary verbs, not copular verbs? For these copular verbs, "all" is an intensifier to the adjective, not a qualifier on the pronoun:
- (wrong) We smelled all good.
- (wrong) We went all crazy.
For auxiliary verbs:
All of the sentences in your example (can/could/may/might/should/must/will/would/ought to/have to) are OK. In addition, these are all okay:
- You have/had all been to school.
- You used to all work at the mall.
- You need to all work together.
- You did all arrive on time.
- You had better all arrive on time.
Negations
"All" can't cross a negation or else it changes the meaning.
We all haven't been to school - None of us have been to school.
We haven't all been to school - At least one of us hasn't been to school.
Unacceptable verbs:
- We dare all go skydiving - Ungrammatical
- We would rather all eat lunch now - Implies "...rather than some of us eating lunch now and some later".
The replacements for defective forms could be okay, but it's abnormal.
- (normal) You should all show up tonight.
- (abnormal) You are supposed to all show up tonight.
- (normal) You are all supposed to show up tonight.
You all have to go to school