Can You Do It? - Modal Auxiliary Verbs

Any of Answers 2 - 4 would be OK here; 'Must we ... ?' (Answer 1) sounds as though they are very reluctant and un-generous, even if you have been waiting for ages with your tongue hanging out.
'May we ... ?' is polite, and offers you the chance to refuse ~ also politely : 'Thank you very much, but no, I'm not thirsty at the moment'.
'Might we ... ?' (Answer 3) is less direct; 'Can we .. ?' (Answer 4) is slightly silly, because it really means 'Are we able to?' ~ and they should already know whether they have any drink to offer you, before they start saying anything! (If you said 'Yes please' and they hadn't got any drink, and therefore couldn't serve you, that would be an embarrassing situation ... )

( = 'How are you able to have two standards at once, you hypocrite?' )
An even more pointed way to ask this sort of question is 'How COULD you ... ?', perhaps with the rest of the question unspoken, maybe because it's obvious what's wrong: e.g. someone borrowed a precious piece of clothing from you ... and came back from a party after being sick, or dropping dark-coloured food or drink, all down the front of it.)
This is a rather dramatic form, like you might see on a TV or film or computer screen. It's very useful but very powerful; we recommend you don't use it yourself, unless (or until) you are very confident of the circumstances.

'Must' is best here, because the person speaking feels that the work-gang think they 'have to' dig up the street at that very same time ~ as though they have been waiting specially for the day.

'Will you ... ?' is asking not only whether it will (probably) happen at all, but whether the person really wants to make it happen ~ if they feel strongly enough to make a deliberate effort to write to you.

'Can you ... ?' is the easy way to ask if someone is able to do something: either now ('Can you stand over there please?') or more generally ('Can you ride a horse without falling off?').

There may be plenty of possible things you could try ... but if you don't do the single most important one, you may not succeed. 'Must' is the strongest word here.

This is the sort of question that we ask ourselves, while we are looking for something.
For even more emphasis we could say 'Where CAN (such-&-such a thing) have got to?'.
Or, more mildly, 'Where might it have gone?' or 'Where could it have gone?'
We know of an old lady who, if ever she lost something ~ or just forgot where she had put it herself ~ used to challenge everyone else in the house: 'What have you done with my ... (scissors, or whatever) ?'!

If you use Answer 1, the person could possibly say, 'Yes, I'm afraid I must, because it's part of my job' (or whatever). Answer 2 would also be possible, but it already suggests there could be a reason ... in which case, you are already admitting that they may be in no position to stop being noisy for you.

Even with the sick child, it is still only polite to ask permission.
Good manners are probably fairly similar all over the world, even if they vary in detail from place to place.
A famous bishop of Winchester, back in the 14th century, had as his motto: 'Manners Makyth Man'! ~ or, as we might loosely say today, 'Human civilisation wouldn't start or stay, unless people treat one another decently'.

It may not be possible for the taxi driver to do this, but at least you have asked if he thinks it's worth trying!