Wait ten minutes?  Wait for ten minutes?

For the first set, each would be correct but in different contexts.

The first one would be fine if it were just a regular comment.

Have you been waiting long?

I've been waiting ten minutes, so not too long.

The second one would be used for someone who is upset about their wait.

I am sorry I am late, have you been waiting long?

I've been waiting for ten minutes, and I don't have all day to just sit here. I have things to do!

Of the second set, it would be common to just use the first sentence, as that is the most direct answer.

A-1 is best, but that doesn't mean A-2 is wrong. In the B's, 1 and 3 are best. While 2 and 4 are not really wrong, the repetition of "for" makes them awkward, but, by deleting "for" before "ten", they would be fixed, although then they'd be the same as the other choices.

A-1 and A-2 are both acceptable and pretty much interchangeable.

You can say all of the B-list forms, but B1 or B4 would be favored.

Probably would put a comma in B4, "for him, for ten minutes".  You don't exactly need the "for ten minutes" to make the sentence, so it is an aside, an added bit of info that is useful, perhaps interesting, but not necessary.

The usual form is "wait (duration) (purpose of the wait), so "wait a few minutes for the train to pass."  "Wait ten minutes for him to arrive."

If you want the other to be the direct object, you need to use the verb await; await the other.   I wait (here, myself), or I await him (his arrival).

Wait is an intransitive verb, does not require an object. I am waiting.

Await is a transitive verb that requires an object. I am awaiting the arrival of the train.