How Are We Doing? - Continuous Tenses

This is the correct, classic Present Continuous form; we use it because the process is 'ongoing'.
For a regular repeated action or fact, we would use the Simple Present ('The sun rises every day') instead of the Continuous form.

This usage emphasises the sense that the other person 'is forgetting' just at the moment (perhaps deliberately, and needs reminding).
We could also say 'Perhaps you have forgotten', or even 'Perhaps you had forgotten' (this latter version suggests that they had already dismissed it from their mind at some time in the past). But the Present Continuous form brings us back very clearly to the 'here-and-now' of the outstanding debt.

They had begun to sing before we came out of the door, and continued singing afterwards; so this denotes a lasting, continued action during which one other shorter thing takes place.
Don't forget that 'everyone' is grammatically singular, despite the fact that it really means a large number of people. ('Everyone is ... / has ... / does ... ')

Again, the sunrise is a longer event than the girl singing. She may have sung for a few moments (and happened to be heard, only for a few seconds), while a reasonable sunrise might last twenty or thirty minutes. Therefore 'the sun was rising' (Past Continuous) over a span of time both before and after her song.

This is a Future Continuous example, suggesting that there will not be any occasion (over some long while in time yet to come) when the other person will have need of the papers.
This structure tends to occur when a manager is 'firing' a redundant employee and says 'I'm afraid we won't be needing your services again in the future'. (I.e.: 'there will be no occasion over a long time when we might need you again' ~ hence the Continuous aspect.)
Of course, one could just say 'You won't want ... ' or 'You won't look at ... ' (NB: NOT just 'look', as in Answer 2). But the continuous turn-of-phrase somehow makes the statement broader and more definite.

Answer 2 is correct; Answer 3 would be acceptable, if slightly informal (using the Present to mean the future), but in Britain we spell forms such as 'travelling' with two L's. (The North American English habit seems to be to use only one, but this doesn't make the meaning any less clear.)

There are a number of English verbs that do not have continuous forms. (We can make them up according to the usual rules of formation, of course, as in Answers 1 and 2; and certain non-native speakers from India and elsewhere quite often do; but they don't really exist.) This example brings in four of them. A lot of the others are also to do with state-of-mind or attitude (like, love, forget, remember etc.).
Be careful with the agreements of number: 'the police' are usually treated as plural, but 'group' itself is singular, which is why the back three verbs in Answer 3 were wrong.

We use 'always' to emphasise how often an action is repeated.
It is perhaps a little confusing that 'always' means 'at any or every time', rather than 'in every direction ('way')' or 'by every means'. We fear you will just have to remember that one!

The first verb is a statement of general fact, i.e. that she knows how to play and does this on a regular basis, or could do so more or less any time she were offered the opportunity.
The second verb is a classic use of the Present Continuous to refer to an event in the fairly near future. More correctly we should perhaps say (though few people would, in normal conversation): 'She will be playing ... '.

'One thing was just happening as another thing also happened' : in this case, the second speaker 'was just reading' the final page at the same time as the first speaker asked about the book.
Again, reading the page might take 30 seconds, while the question would be quicker, maybe just a few moments. But the second speaker is keen to emphasise that the reading of the last page was happening right at the same time ('by happy coincidence'!) and is now complete.
Perhaps you were just wondering how many points you scored on this Quiz? ('I was just wondering / looking' etc. is how someone might say that they weren't really interested in something, when perhaps they actually were; 'I was just saying' could mean either that you were mentioning the same subject a little while ago, or that you were 'just' (= 'only') saying something, maybe pointing out an awkward detail in someone's plan that you felt ought not to be overlooked or forgotten.