As I Was Saying - Simple & Continuous Tenses

'Remember' does not usually have a Continuous form, although on certain occasions someone might say (e.g.) 'Please excuse me if I look a bit sad this evening, but I am remembering my granny who would have been 100 years old today.'
In this case, 'You do ... don't you?' is a classic emphatic tag-question to 'jog someone's memory'.
We use 'are coming' to refer to the fairly near future, in the sense that your aunt and uncle themselves are aware of the visit too, are actively bearing it in mind, and may even have begun packing and preparing already (so the whole 'cycle' of the visit-event is underway; hence the Present Continuous).

Even though this poem is nonsensical, the language of it works as usual ~ else we readers (English or not) would be lost, and give up!
'The sun was shining .. ' is a long(-ish)-term, continuous happening (even in Britain, if you're lucky with the weather!).
'The middle of the night' (i.e. midnight) is a relatively short moment in time. In any case we do not usually say 'was being' (as in Answers 2 & 4) except when we are forming a Past Passive ('I did not realise the road was being repaired, or I would have come round another way').

These are quite complex and subtle verb forms, but it is very useful for you to see a sequence like this and think through how it fits together.
'All being well' can be tacked onto almost anything, somewhat like 'inshallah' in the Muslim world: it means 'if all is well (as we pray it may be)'. Usually it refers to a present or future action when we can't be directly sure about it ('All being well, his plane should be in Cairo by now').
The middle verb-group is a Future Perfect, describing action which 'will have' been finished by some point in the future. Often it will be a process that has already started by the time when it is being discussed ('You will have finished this Quiz in a few more minutes').
The last group is Future Continuous: the action hasn't started yet, but at our chosen point in the future it will already be underway and carry on further beyond that time (hence, the Continuous element: -ING).

Check each of these usages carefully, by way of revision:
The first part would normally be in the Present Continuous but suggesting the future (with the idea of 'while you are, or will be, travelling': this is a process that necessarily has a certain length to it!).
There is no need or cause for a continuous form of the middle verb, which is a single quick action that someone either will do, or they won't.
We would say 'it is fun doing something' or 'fun to do': 'It would be scary walking through the forest at night'; 'It is exciting to watch them playing'.

Answer 2 is also possible as a shorter, conversational version; Answer 4 is possible, but perhaps over-detailed and deilberate and clumsy (though there is a serious decision, and serious reasoning, involved).
The main point here, from the language point of view, is the use of Future Continuous forms : 'I shall avoid, and continue to avoid ... '. Of course, English often then 'avoids do-ING' an action rather than just avoiding a thing (such as snakes): 'Try to avoid walking in those muddy puddles!'.

'Your' trip to St Ives would have taken a certain amount of time ('continuous'), but if you 'met' the other person (or people), that would have been a shorter event ~ particularly if they passed you briefly on their way in the opposite direction. So we have one longer action, and another quicker one during it.

These are both straight 'factual' verbs, and neither of them properly has a Continuous form anyway.

The singer describes that he 'already is leaning' (i.e. he was in that position since before you saw him). If the lady does come by (= walk past), that will be a single and shorter action, so we use the Simple Present. He may very likely be waiting a good long time 'on the off-chance' that she may come along!
Incidentally, the show title 'Me and My Girl' isn't quite 'proper English', because the man ('me') is a poor young man called Bill Snibson. He has been brought up in the East End of London and does not speak standard ('educated') English, where the correct phrase ~ as you probably knew! ~ would be 'My girl and I' (like the Queen with the Duke of Edinburgh 'My husband and I')! In the story he suddenly comes into a large amount of money and a big house in the country, and much of the comedy in the show is about the differences in behaviour and language between how he grew up and what is newly expected of him. This is also a leading theme in another famous musical, 'My Fair Lady'. The English aren't actually obsessed with this issue, but it makes for an interesting story (and lots of good songs)!

Answer 2 includes the Future Continuous: suggesting that, at some point during a length of time, the other person may (or may not) have a sudden or definite need to use the car.
'Do you need ... ?' (Answer 1) is a simpler and just-as-effective way to ask the same thing.
Answer 3 is possible, in the half-expressed sense of 'Were you expecting to be needing ...? Were you going to be at all likely to want ...?'. A phrase like this would probably sound very odd coming from anyone other than a native speaker of English.

'Are you going ... ?' means 'Are you already going, or just about to go (in a few short moments from now)?'
'Wonder' is another verb that rarely, if ever, forms a Continuous version. (We do sometimes say 'I was just wondering ... ' when an unresolved idea has come up again in the back of our mind; but 'I wonder' is the usual simple present form.)
We use 'you are' (Simple Present) to refer to the future time when the other person will be in the town. This may not be absolutely logical, but it's 'quick and easy and simple', not hard to make or understand, and quite a lot of other languages do the same thing. (Does yours, we wonder?)