These questions of ours

Books are evidently countable objects, so Answer 3 is a non-starter (though with an uncountable commodity such as 'cider' or 'paper', it would have worked, at least that far in the sentence!).
Meanwhile you would not ask permission to borrow anything except from its owner, so Answer 4 has to be right as it is the only one referring to 'yours'.
The books are more likely to be 'these' (that we've picked up, and like the look of) than 'those' (further away; less attractive or involved).

The context suggests continual and/or repeated 'attacks' of whatever the medical condition is, so clearly it should be plural, which rules out Answers 1 and 2. Answer 4 may seem more compact and appealing, but that is not the way English expresses such ideas (cf. the Quiz title)!

'Any old friend of mine' (Answer 1) sounds like a too-convenient excuse for delay, any very probably not even true!
'Any' in Answer 4 makes equally little sense; nor do we know who the 'he' is (as in 'his': Answers 2 and 4).
This leaves Answer 3 ~ where the 'some old friend of yours' is better-known to the person being spoken to, than to the speaker (who apparently can't remember this person's name).

Given that a daughter is a human being, 'what' would be inappropriate; 'which' is better.
Answer 4 would have been acceptable had it said 'Which of his daughterS ... ' (i.e. which one amongst the two-or-more).
The accidental juxtaposition of ' ... his is it ... ' may look and sound strange, but in context is entirely correct; some writers would perhaps put a comma after 'it', to clarify the boundary between the component clauses in the sentence.

You may still find it strange (even if your own first language is very precise with its grammatical endings and other markers) that we say 'of theirs' (Answer 1: correct) rather than the grammatically plausible version at Answer 2, but Answer 3 is a step too far away, in that 'of they' really doesn't work.
The version at Answer 4 would certainly be understood, and might even be a tidier way of expressing it ~ but nevertheless, it isn't what we say.

'What' (Answers 1 & 3) does not apply to a person ~ even a daughter's long-abandoned boyfriend!
It is not a matter of 'whose' (Answer 4) former boyfriend this was; the idea of 'ownership' or origin ('from what boyfriend') cannot accurately be expressed this way.

The customers must be outside the shop, else they could already see whatever the surprise is going to be; therefore they must be (to some grammatical extent) 'distant', i.e. 'those' rather than 'these'. This leaves the middle two Answers, of which only No.3 is correctly expressed.

' ... Any such behaviour of his ... ' is the clear nub of this sentence.
The context, as you may have picked up from various clues, is a formal letter from the headteacher of a school to the parents of a student who has been in serious, perhaps repeated trouble.

The children and their bikes are at least some short distance away (we can almost sense the pointing finger!): only Answer 4 fits this situation, both geometrically and for good grammar.

'This' suggests that the conversation has moved onto the topic of the boyfriend, even though ~ presumably ~ he is not physically present (as in 'this is him: here he is').
There is only one letter of difference between Answers 3 and 4, and 'we' are clearly keen to hear about the young man, but Answer 4 really does not make such plausible sense!