Have we done it?

Answer 2 is the most fluent in such a spoken context. The form without the auxiliary 'have' (see variants at Answers 1 & 3) is more an American usage, and perhaps also specifically Jewish (try reading them aloud, or in your head, in a Woody Allen voice ... except you'd also need to say 'trash' instead of 'rubbish'!). Answer 4 is fine, but perhaps a bit too formal or literary for such an everyday (or every-week!) topic.

'If they had realised, they would not have done it.'
' ... is' (Answer 2) is slightly better than '... was' (Answer 3), because asbestos remains dangerous even nowadays, and even if it's all removed and ceases to exist, the substance still retains that theoretical property. So the present tense remains 'universally true'.
Any Answer including 'shouldn't have' will carry more of a moral force: 'they wouldn't have ...' suggests that the builders can't be faulted for not knowing what they couldn't then have known (what Donald Rumsfeld would have called 'unknown unknowns'); but 'shouldn't have' suggests that they ought not even to have started, whether or not they were sure.

We certainly do need the Pluperfect sense of 'come' here, because how her hair appeared was (presumably) a result of what had happened to her at a point further back into the past.
None of these other forms is appropriate here, though they would probably be understood.

The two Present forms (Answers 1 & 2) are possible in context, but do reflect the usage in many other languages ~ certainly both north and southern European languages ~ which say 'someone does something since a point in the past', because the action is still ongoing and therefore justifies the familiar present-tense form. English prefers the past continuous ('I have been doing jigsaws since I was young').
Likewise, out of context there seems little justification for the emphatic/continuous form in Answer 4 (although there's a slight sense of 'grudging', perhaps from the parent/s who bought him the original instrument and paid for lessons, as though to say 'Well, at least the early start gave him a bit of technique, and let's be glad he is still actually making some use of that instead of dropping it completely'!)

Past continuous / progressive is the most elegant (and eloquent) tense here: it suggests that the extra journeys are typical, and cumulative, and they seem to be as bad as ever for the moment, but with a faint hope that they may improve at some point in the future when the system is put back as it was before (e.g. as with temporary roadworks).
None of the other Answers is entirely wrong (even the present-tense ones, and despite this being a Quiz based on Past verb forms); but No.3 is clearly the most accurate and vivid.

Answer 2 is also possible; Answer 1 would only work if he (or, indeed she) were still living locally, i.e. if the action ~ in this case, playing ~ were still ongoing.
Answer 3 is possible too, in the sense that this was an ongoing and frequent habitual part of his life back then, but perhaps now his health and/or enthusiasm have declined and he would not be able to 'get back into the swing' in quite the old way.
Answer 4 makes it clearest both that this was a habit, and that it is now over.

The simple past form is best for an over-and-done-with, one-off historical event.

('I wasn't aware that she DID ... '); Answer 4 is also possible, since ~ presumably ~ 'she' is still just as much a Jamaican as she ever was.
Answer 1 (in the Pluperfect, or Past Perfect) suggests that she had come from Jamaica ~ prior to when the speaker met her ~ and that she had had something special done to her hair while she was there.
Answer 2 conveys a sense of the Imperfect, but it's wrong in this situation, suggesting only that she was somewhere along the middle of her journey away from Jamaica and maybe something had happened to her hair during that trip. This doesn't make very plausible sense, does it?

Answer 2 is fair enough, but Answer 3 with its contrasting tenses (birth being more or less a single event, but upbringing taking noticeably longer) is even better stylistically.
There are mis-formations in Answer 1 ('birthed' is clear enough but wrong; 'bought' is to do with Buying rather than Bringing ... don't let's explore that any further!) and Answer 4 (with no need for the Pluperfect).

('I have had to speak to you ...')
Answer 2 is more a North American usage; Answer 3 is simply, though understandably, wrong (with its Imperfect / continuous form suggesting that such disciplinary meetings were indeed a fairly regular occurrence); Answer 4 is just about possible, but sounds very mannered and old-fashioned ~ no modern employee would take such a usage as seriously as the speaker would have meant or liked!