A test of your skill

English can stack three or more nouns into a phrase, to make a clear but concise formation as in Answer 2: the souvenir is from a rowing activity within the college that Uncle Theo had attended.
It seems unlikely that the souvenir was 'theirs' rather than Uncle Theo's (at least, at first-hand), which eliminates Answer 1; Answer 3 is possible, but somehow 'from college' (or 'from his college days') would seem more natural than what was offered.
In Answer 4 there seems an untidy clash between 'from rowing' and 'of the college'; a souvenir 'from ... ' would, in any case, more usually indicate a place of origin, rather than an activity, and 'a souvenir of rowing from the college' doesn't work very tidily either.

The key element here is to identify 'the case' as the focus of such ardent attention of the auction. Working outwards from there, we establish that it was a cigarette case, rather than a hatbox / suitcase / pencil-case, or whatever else; then the material from which it was made, goes in front of that to give us the phrase 'snakeskin cigarette-case holder'.
The hyphen is merely a precaution in case anyone should think that the cigarettes were made of snakeskin (as in Answer 3) ... we wonder whether even someone in the Gatsby era would have compounded their vices with such determined extravagance! (i.e., not only wrecking their health by smoking at all, however fashionable; but by using materials made from the remains of an endangered species!) ... and one dreads to think what burning snakeskin would do to the flavour of the smoke ...
Answer 1 is quite a good shot, but not fully natural; the 'old snakeskin' (in Answer 4) seems slightly unnecessary, in that the case itself (and the material with which it is faced) is already 'vintage', and equally obviously, the former owner of the skin (i.e. the living snake) lies even further back in the past, so 'old' in this position adds little to the meaning.

The helpful object, as we know from the introduction, must be suitable for the purpose (i.e., in this case, something to sit on) and also belong to the person who is being reminded about it. Such an idea would usually be introduced by a 'relative' word ('this'/'that'/'those' etc.), so Answer 1 is a non-starter.
Answer 2 is fine and suitable, though you may wish to check 'shooting (-) stick' in a good dictionary. It refers to a stick that converts into a seat (of sorts, at least propped into the ground to bear one's weight); such as one might use while out on a shooting-party ~ rather than that the stick, itself, were any kind of a weapon!
Answer 3 introduces 'their'; who are they? This doesn't quite make self-sufficient sense.
'Walking boots' (Answer 4) might help the problem of comfort in the open air, but hardly very directly.

Let's get the jacket sorted out first, and leave the explanatory ownership reference to last.
The jacket was of a style / purpose to be 'a smoking jacket', made of tuxedo in an ivory colour (light-creamy, elegant off-white); so the version in Answer 4 conveys that clearly and in the order an English listener or reader would expect.
Answer 1 appears to suggest that the ivory of which the jacket was made, was itself smoking (ivory being the solid substance of which elephant tusks are formed; so a solid inflexible jacket, which was also giving off smoke, would be almost as impractical and alarming to wear to a party as Cinderella's 'glass slippers' (actually less likely 'verre' in French, than 'vair' = squirrel-fur: far more supple and comfortable!).
The remaining wrong Answers, though 'trying hard', are equally messy and confusing.
Several languages have quite specific rules and hierarchies about where broadly comparable elements should go in order within a sentence. German, as you may know, arranges any two-part verb with the front part as second 'idea' in the sentence, and the other part sent to the back; in between, there may come details of 'time, manner and place' in that order and no other, such as:
'My aunt and uncle have last year to Spain by steamer cruised.'
(Any other order might well stand a chance of being understood, if you were thinking it up as you went along; but it would be 'technically wrong'.)
So, you see ... 'it's not just us'! ... and indeed, having rules and procedures actually helps make the 'communication game' that little bit more standardised and easy to manage.

In English, when we have a transactional verb with two Objects, we 'buy a thing for someone' or else 'buy someone a thing'.
Answer 1 is understandable but unnatural in English; the preposition 'to' is also understandable, but unnatural, in Answer 2; in Answer 3 the adjective has ended up in the wrong place.

Many of the possible Answers here are reasonably clear and natural, but No.1 is the best.
The 'oldness' of the father (Answer 2) seems an unnecessary emphasis, unless it is an 'endearing' usage.
Answer 3 contains a series of clumsy possessive constructions ~ and would only appeal to speakers of certain Mediterranean languages, that have to use prepositions instead of cases, to express ownership.
Answer 4 puts matters in rather a different way, with the supposition of 'partnership' being new, so it does not quite bear comparison with the others.

They are fellow-singers of long standing (i.e. they and we have been making music together for a considerable period of time, such as in rehearsals and concerts over a number of years and works); this does not mean that they are used to standing up on their feet for long whiles at a stretch (although choral singers usually do have to get used to doing that)! The fact that they are our own friends, which in the context we can almost guess may be coming, can be left to last to say.
The version in Answer 1 seems almost to suggest that 'they' and 'we' are the only performers ~ which is possible, but unlikely within a larger ensemble. If the whole ensemble just consisted of two couples or even families, the apparent introduction would perhaps be unnecessary.
'Our long standing' (Answer 3) is an unhelpful mis-coupling of ideas; Answer 4 is the least readily understandable of all.

Answer 2 is both the most concise and elegant; a 'book of sketches by me' (Answer 1; like 'a concert of pieces by me') sounds like a contorted effort to be modest and distance the speaker from his/her own work; and that, in turn, makes poor sense in the apparent context ... if the speaker is actually offering the book around for others to look at (and, presumably, express positive interest in ~ if only for the scenes, rather than to applaud the artistry).
Answers 3 & 4 are each quite seriously malformed in terms of grammatical organisation.

Answer 1 would be clear and possible but slightly unnatural; the version in Answer 3 is better (and notice the apparent double possession indicators in the phrase 'of your grandad's').
The order of ideas in Answer 2 seemingly confirms that 'grandad' is 'old', which outs the emphasis on age in a less relevant place; the order in Answer 4 is fairly clearly silly.

The underlying sense of the sentence is: ' bar = place where they drank ' ; the adjective needs to slip in immediately after the 'equals', in front of the 'place'. This leads us to Answer 2, where the relative [!] detail that the bar was favoured by his parents, belongs beyond and outside this structure.
Once we have created, or recognised, the key phrase 'favourite drinking (-) place', we can discard Answers 1, 3 and 4.
Answer 4 also raises the ticklish issue of whether 'parents' should carry an apostrophe, and if so, where:
'His parents drinking' (as in Answer 4) makes sense but is ungrammatical.
'His parents' drinking' = the drinking that was done by his parents (e.g. what they drank; whether too much, e.g. so as to become a problem for him; or the behaviour as such, i.e. that they did a lot of it, or indeed how or where).
'His parent's drinking' suggests that one of them (singular, as noted form the placing of the apostrophe) was an alcoholic, without even being firm enough to specify which parent.