Unless you object

... i.e., 'Only come in if you are prepared to face arrest; else stay out'.
None of the other Answers makes plausible sense in the circumstances; check them again!

A preposition is clearly called for in front of the noun phrase here; not a conjunction, or anything else. (Any other preposition, such as 'without', would fit the grammatical context ~ though that wouldn't make a lot of sense!)

'While' (Answer 3) is right; we could also have offered 'but' or 'although'.
'While' suggests that the two actions took place simultaneously, i.e. the two brothers were working on opposite sides of the world (apart from the fact that during the waking/working hours of either one of them, the other would have been asleep, because of the time-zone difference!).
One needs to be careful about sloppy use of 'while', though ... see our next Question for more on that!

'While' (Answer 1) again implies two contrasting things happening at one and the same time: had both pieces been performed at once, the combined sound would probably have been excruciatingly disturbing! Dear old 'And' (Answer 2) very simply bypasses this unfortunate suggestion.
The other two Answers (3 & 4) raise hints of rivalry between the girls ('So which is the better musician?') ... rather lacking in 'harmony', dare we suggest, and certainly at odds with the whole idea of collaborative music-making for a good cause (in the context of a charity fund-raising concert). Incidentally, you may have spotted that it should have been " 'Memory' from 'Cats' ", rather than " 'Cats' from memory " ... although there are seemingly innumerable sopranos (and others) who will sing this famous Lloyd Webber song 'out of their head', given the least opportunity!

'And' does not suggest any contrast between his good and bad days; 'despite' (Answer 3) is a preposition, so it should not introduce a clause; 'nonetheless' is a good conjunction, but we were already given a comma before the blank, and this would have needed a stronger punctuation mark, almost certainly a semi-colon (';'). One does see main clauses separated only by a comma in what ought to be decent respectable English, but ~ except in fairly rare circumstances ~ that still isn't technically right. Where the second clause amplifies or explains the first, a colon is a good separator ('Let's go through to the dining room: we can continue our conversation over supper.'). Indeed, the very symmetrical nature of the two dots as your eye passes onwards along the line feels rather like the dinner guests walking through the doorway into an adjoining room, which is connected and yet intentionally distinct. The semi-colon, on the other hand, is asymmetrical ('wonky') ~ which suggests that the second part of the sentence will still be related, but go off on a contrasting 'diagonal tack': 'The management on this project has been questionable; but our frontline staff have continued to do a magnificent job.'

The only completely wrong Answer here is No.4, which really fails to make much sense.
'And' (Answer 1) is possible but really rather bland; our old friend 'while' (Answer 2) is actually all right here because despite the contrast in running sides, it may be happening simultaneously but in different places.
(Readers who have come from overseas to Britain may recall the experience of arriving and adapting, but at least you would have had to cross the Channel by non-driving means between driving abroad and here, e.g. by ferry or flying; crossing the land border into / out of Hong Kong and elsewhere feels far stranger, as within a matter of minutes you are on 'a patch of land just next door' yet where everything on the streets is the other way round!)

Each alternative here, other than No.3, expresses the contrast between the adverse state of 'his' health and the diligence with which he fulfilled his duties or expectations.

Some people might say 'regardless' ( = 'irrespectively') ~ without hardly realising that even in this context, it suggests a lack of esteem ('holding something in scant regard') which seemingly undermines the message about respecting 'him' for his hard work.
Note the small ~ but not insignificant ~ difference in nuance between 'none the less grateful' (Answer 1; i.e., 'Please don't think I am thanking you any the less sincerely, just because there was a misunderstanding about the urgency') and 'nonetheless grateful' (Answer 3: 'I remain just as grateful as I would have been, had we in fact needed that report by today').

Answer 1 is the weak one here; what's the point in going out 'until' one finishes one's homework? (Would you take it out with you, and come back as soon as you'd finished it? Only, perhaps, if you were sitting outside in the garden or park with it on a summer's evening, which we suppose would make possible ~ if unlikely ~ sense!)
Answer 4 is at least equally possible, but on a slightly different tack, as the homework is apparently already finished (rather than being an unfulfilled condition for later on).

In other words, the rest of the team will carry the match ('despite him') instead of him being the star scorer.
Answer 1 has the right expressions but the wrong way round, so it contradicts the sense of how the sentence was set up.
In Answer 2, 'because him' needs an 'of' in the middle; we might, we suppose, have 're-factorised' the parallel expressions as ' ... in spite, rather than because of him', except that the now rather disconnected phrase 'in spite' raises the awkward suggestion of further dissent within the team (i.e. someone was mis-playing out of spite against someone else within the side).
Answer 4 commits the reverse mistake of including a redundant 'of' after 'despite'. Remember that 'despite' = 'in spite of'.