If this, then that

This is another way of saying that if a potential job or responsibility is unawarded or unallocated, but you have the right skills, you should think about taking it on yourself. (Perhaps the thought was of an old-fashioned official cap, such as used to be worn by various officials e.g. tram conductors and park-keepers.)
This is nothing to do with the 'cap' as in Answers 3 & 4 ~ which consists of a small quantity of gunpowder (or similar) in a paper wrap, which would make a sharp noise if suddenly hit by the mechanism inside a toy pistol.

This is the only correct use of the previous past conditional form, though it could equally well be written ' ... never would have gone ...' (a version which wasn't offered here, as we are only supposed to provide one good/best Answer!).

This is definite enough that only 'will' could be sufficient; the other Answers are all too feeble in their meaning. Why put up a sign at all, unless there's definitely something to tell people?

Answer 1 makes grammatical sense, but in terms of meaning it is 'inside-out': if there WERE a problem, they surely WOULD call you as soon as they were aware of it.
The other two links are wrong: 'until' suggests that the medics would, perhaps deliberately, delay calling you until the problem was serious enough (instead of alerting you earlier, so as to stand some chance of halting its development); 'except' is a preposition rather than a conjunction, so it isn't strong enough to introduce a complete clause rather than just a noun-phrase.

'Because' (Answer 1) seems to be following through on a previous communication from you, which ~ out of context ~ we don't know, or can't be sure about; 'whenever' (Answer 3) suggests an ongoing dialogue with the company (which, surely, neither party would really want to have to enter into); 'although' (Answer 4) is also clearly the wrong conjunction.

This is another, more polished way of saying 'If there had been a problem, she would have told me'.
Answers 1 & 3 contain false constructions instead of 'there is/was/were'; if Answer 4 were to be correct, it should have run 'In case there WERE ... ' .

Answer 1 is a good, plain way of putting this; Answer 2 is quite unsuitably pompous, while Answers 3 & 4 (each, phrases you may well hear spoken) seem too informal after the official start to the sentence. Presumably we are discussing the potential safety of a trip in difficult conditions, so a reasonably serious tone would be appropriate.

Answer 1 would have worked well had it said 'wheNever' (which it didn't; look carefully!). The 'should' in Answer 2 would have led into ' ... should the statistics become available' (which, again, is not what the Question offered). If the figures ARE already available, 'given that' (Answer 3) seems very churlish; the authorities seem to be saying, 'Since we happen to have the facts and figures, we might as well let you know ... ' ~ which hardly seems a model of open, proactive customer-facing authority.
'As and when' (Answer 4) concisely expresses that the material will indeed be released in the manner that it says, i.e. not yet, but as soon as it is ready; a very handy phrase to recognise and use situationally.

Answer 1 suggests that the mobility problems make the people appreciate the club, which is certainly possible but not necessarily likely; 'although' is a conjunction rather than a preposition (cf. the reverse situation in Question 5, Answer 3); 'nonetheless' (Answer 4) usually stands by itself. The most likely sense of the sentence is that the old people enjoy the club EVEN THOUGH it is an increasing effort and upheaval for them to get themselves there.

Either version of Answer 1 is fine; Answer 2 is possible, but pedantic to the point of being inappropriate; Answer 3 contains many good elements but is too garbled to do the job clearly; Answer 4 is theoretically acceptable, but a native speaker would assume from the high-flown style that you were trying to make some cynical point (it sounds almost as though a knight from mediaeval times is tackling his 'chivalric ordeal' on a prime-time television trivia quiz show!).
This leaves Answer 1, at the top, as the best possible option.