What will you be doing next?

The first element of this situation uses an actual Present (continuous) form to refer to a single future event. Many other European languages make a similar, simpler substitution like this, but if your own language takes such plans more literally, you may need to continue to adapt!
The second element (complete with contraction) is a very natural way to express a potential future event that is much less certain, and probably not 'fixed' ~ certainly in the current speaker's mind ~ in the same way as the front part of the situation. We use this form (usually contracted) when predicting, from past experience ~ as here ~ how matters may develop. ('If I know Uncle Charles, he'll have had a drink or two before he gets here.')

We usually use the Present form in an 'if'--clause, even though this refers to a future condition which may yet arise.
The form within the indirect speech (2nd line of the Question) suggests that the speaker/singer WILL be on his/her way by the time that dialogue happens, but that hasn't started yet either.
So we have a classic pair of lines referring to a (likely) future event, yet without any obviously apparent Future formations.
Most of the other alternative Answers offered here don't fit the rhythm of the song, either!

' ... Am going to be ... ' is the appropriate phrase here, but we hope you weren't caught-out by the sneaky extra English catch in Answer 2. People's present (or, in this case, future) jobs are almost always introduced in English by the Indefinite Article (as in 'My old man's A dustman ...') ~ hence, Answer 3 is the better version.
Answer 4 'means well' ~ but is adrift of what we'd actually say, in respect of each of the two word elements.

We have two Future elements here: 1. that the speaker will answer the phone him/herself; 2. who will turn out to be the caller.
Answer 2 meets both these requirements best. The 'shall' in Answer 1 is inappropriately over-determined, and 'respond' (though fully understandable) is not how we would express this (Answer 4).
The Future form in the second section of Answer 2 expresses the idea that once the call is answered (which hasn't quite happened yet, and is therefore Future), it WILL then turn out to have been Auntie Fleur at the other end of the line.

The birth of the baby is increasingly imminent, and represents (shall we say?) the culmination of an ongoing ~ Continuous ~ process, as much as the starting of the even greater process of increasingly independent life. So a continuous verb-form (as in Answer 2 or 3) seems particularly suitable. The auxiliaries in Answers 3 and 4 are too strong though (something might, sadly, go wrong between now and then, so childbirth isn't quite 100% inevitable or compulsory).
Although the baby belongs to both of 'them', we usually speak of a woman 'having A baby' ~ come to think of it, that indefinite article does seem a bit surprising, when she has been making that baby her own for 3/4 of a year ~ rather than 'her' or 'their' baby or even just 'the' baby (grammatically Definite since we all know it's on the way, because we can see and/or have already been told about it).
One way and another, Answer 2 is the best here. Meanwhile the 'thin chances' in the Question are perhaps a slip of the speaker's tongue (or mind) ~ since Megan herself will, presumably, be growing larger over the months to come!

(Please note that this is/was a fictitious example, and the carefully-specified date!)
Even the Government ~ any government, at least in 21st-centurty Britain ~ should be unlikely to declare such a development as boldly as in Answers 1-3, which seem to be increasingly petulant.
The 'correct' and appropriate way of phrasing this, as in Answer 4, is still fairly definitive and resolutely monosyllabic. This is how we (or anyone) can make statements about matters that are officially planned, whether or not we personally approve of them: 'Doctors are to be given access to a new all-singing, all-dancing IT system', etc.

Answer 2, in turn, presupposes such a reply as : 'Oh yes, of course, I will'.
Answer 1 would be possible (though still rather blunt) even if we removed the over-inquisitive 'why' from the front of it.
The 'shan't' form in Answer 3 is not appropriate; Answer 4 is just about possible, but sounds more forceful than friendly, and so hardly belongs within the original context of such a dialogue.

Future continuous matches this circumstance the best.
We do sometimes use a simple present verb to express future action ~ but not with weather conditions, which typically last for a while (hence the Continuous element in the language to say so).
Even in Britain, rain is still not quite compulsory (Answer 3)!

Answer 3 is best here because the negative question is a firm reminder, expecting such a response as 'Oh yes, so she is'.
The form using a past tense (Answer 4) is possible, and might certainly be heard, but the very past-ness of it suggests that this idea has now been superseded; i.e. someone has changed their mind since that plan was proposed before.
The affirmative question in Answer 2 does not work particularly well here; meanwhile, the version at Answer 1 is a statement, rather than being a question at all.

This is a case of 'the future in the past': at the time to which we are referring back, the expiry of the insurance would still lie in the fairly immediate future: the 'going to' structure works better than any of these others in this context.
Any of the other Answers would be adequate and acceptable, but No.3 'has that edge'.