Between ourselves ...

... 'at a time, on a date'; in British speech we would actually say 'on the 22nd of November'.

We pay a bill (etc) in English; but we pay it, by means of money or a card or whatever, FOR an item or service, i.e. in exchange for it.
We pay a person for an item or service; and if they did the original transaction on our behalf, we pay them back for it. The phrase 'pay back' can also be used of someone seeking revenge (not necessarily in terms of money): 'I'll pay you back for damaging my precious car'.
Please do have a care with verbs taking two objects (direct and indirect): 'I bought a hat for my sister, and while I was about it, I bought myself a scarf' ... i.e. you buy a thing (direct object) FOR someone (indirect recipient), or you send it TO them or you shout it AT them; but you buy somebody a thing, tell someone a secret, promise them a reward etc., and if you do it this way round, with the person specified first, you do NOT need any preposition. Coming from various 'other' languages, that may feel strange ... try treating yourself to a little further practice!

Answer 1 is the best; if you are uncomfortable with the two idiomatic 'ats', you can drop the first one right out without affecting the meaning too much (although there's still a worthwhile, indeed life-and-death distinction between them 'shooting at you' by way of a general warning, and 'shooting you' direct, deliberately and possibly with specific intent to kill or at least significantly injure you).
Smiling 'for' the President is a fairly good alternative (Answer 2), but the remaining two Answers ~ though generally understandable ~ do not qualify as clear standard English.

All these Answers apart from No.3 contain errors of logic or style, or both.

Answer 3 is the best; most of the rest are understandable but wrong. 'Taking back' the farm (Answer 2) is acceptable, as you may recognise if you know the story, but only on the assumption that it should have been fully theirs to begin with.
'Animal Farm' is a good read (and not too long; around 100 pages in a typical standard paperback edition), which we would recommend for all sorts of reasons: a fairly recent classic novel (though Orwell titles it as 'a fairy story'), told in consistently good English, with a strong socio-political point, but which can be read with enjoyment on a variety of 'levels'.
If your own country happens to have any totalitarianism in its present system, or within living-memory history, you will find it especially interesting.

In English one looks FOR something that has yet to be found; and looks AFTER a thing or person entrusted to one's care ('You can borrow this book, but please look after it carefully'; 'This is Mr Snodgrass who looks after our technical department').

Answer 4 is the best pair, driven by the final idea of 'talking AT' people (which suggests that the speaker is delivering what they planned to say, completely regardless of what response they are getting). Answer 3 tries to convey a rather similar contrast, but somehow 'doesn't quite work' as an expression.
Answer 1 is possible but probably less likely (the idea being that such a teacher would converse with you face-to-face rather than discussing you 'behind your back'); Answer 2 suggests one who cuts in and interrupts you, rather than waiting until you've had your say before commenting on it.

We congratulate, applaud (etc.) people ON ~ or occasionally, 'for' ~ being good AT something.

Answer 4 'rings all the right bells' here.
'He' can't sit 'among' his parents since (at least biologically) he can only have two of them, so the correct preposition is 'between' (as in 'deciding between two options, in a dilemma'). The corollary of this is that one can't be 'between' a crowd, what with a crown needing to consist of rather more than just two people.
Elements of some of the other Answers are promising, but no other complete Answer is adequate; if 'he' were 'sitting behind' his parents (Answer 1) from the photographer's viewpoint, we wouldn't be able to see him in the picture, while 'amidst' is an unnecessarily old-fashioned, pedantic word that sits ill in this context.

The two different prepositional idioms with 'looking' are the nub of this Answer; the 'identical' versions at Answer 2 are appealing, but the pun does not work entirely naturally.
In Answer 1, 'dealing with' would have been acceptable; Answer 4's double use of 'watching' does not quite work, because she would (probably, more likely) have been watching outside, or through, the window rather than watching the window itself. (One can 'watch the door', i.e. 'keep an eye in case anyone comes', and famously, on the London Underground and elsewhere one is encouraged to 'Mind The Gap'.) One might also 'watch over' a sleeping child, pet or someone that was unwell (like the shepherds on the hills in the Christmas story), but 'watching over' a range of documents does not quite feel natural or appropriate.