Learn English – What are “modal-verbs”

Both of the past-tense examples sound somewhat archaic, but that is to some extent because the use of shall in the present-tense sentences does also, imo. (The use of present-tense may in the second sentence sounds formal, but not archaic.)

Probably the last person I heard use might and should with these specific meanings was my grandmother, who learned English as a foreign language about 100 years ago (literally). If your questions is whether you should :-) use the words with these meanings, I would say no, as it will simply be confusing. To convey the past-tense sense of these, you might have to come up with workarounds:

When he was at school, he was not allowed to go to the bathroom ...

He thought that they would go to the restaurant.

Hmm, that second one is tricky, but it's the result of substituting the non-archaic will for shall in the present tense.

In your list, only the various forms of be (the first eight) and become, seem, appear, look, smell, taste, sound and feel are copular verbs. Others include remain, keep, stay, get, go, come, grow, prove, turn, turn out, end up and wind up.