Easy as pie!

Most of the others are fair phrases in other contexts, but only Answer 3 is right here; and what could be more British to start us off?

'Last gasp' (Answer 1) suggests he is about to die, perhaps of a breathing problem ~ this can also be used metaphorically, e.g. of the final steaming of a railway engine or power station, factory chimney etc. (some machine that looks as though it has been breathing while it was still working).
The last lap (Answer 2) is an image from the world of racing; depending on what activity 'the club' is about, it might be fairly appropriate here. The 'final furlong' is certainly the last length run in a horse race.
A 'final fling', complete with the benefit of alliteration, means someone will be 'flinging' (= cheerfully, energetically throwing) themself into something for the very last time. Sometimes it also refers to a man (or perhaps less likely, a woman) having a brief relationship with someone else before taking their marriage vows.

Answer 4 suggests the idea of drowning (about as serious as 'losing one's head', though also ~ presumably ~ in a metaphorical sense). Answer 1 suggests he talks a lot of nonsense, which would presumably waste time without helping matters along; in Answer 2 he would not know the front end of an idea from its back (as in the image of an animal: not much use when dealing with livestock of any kind, or the metaphorical 'snakes' of enemies!); Answer 3 could also be rendered as 'away with the fairies', i.e. unable to concentrate pragmatically nor consistently on the matter in hand.

The first three Answers are all expressions of chaos and lack of control, from the worlds of international trade (Answer 1), the London guilds and 'livery companies' trying to solve a dispute over their order of precedence in formal processions (Answer 2), and the retail ~ or possibly, manufacturing ~ trade (Answer 3).
Also from the world of shipping ... Britain was, and is, an island nation, don't forget ... comes the image at Answer 4 of a ship reaching port safely, with no leaking water to spoil her cargo.

'Touch and go' seems to suggest some form of outdoor game where the player touches something and then has to go away and wait. This is a phrase we use in such circumstances as this Question, but we could easily also have offered 'wait and see', 'hit or miss' or indeed 'as and when' (this latter, suggesting that the very idea of a deadline is purely notional and almost infinitely elastic!).
All the others are good, useful orthodox phrases but none of them fits these circumstances so well.

'Over the moon' (Answer 1) = overjoyed, psychologically as 'high' as anyone can feel; 'Over the rainbow' is a quotation from a song in the musical 'The Wizard of Oz', but not particularly appropriate here; 'over the hill' is a slangy and rather disrespectful expression for a person (or, less likely, a thing) that is old enough to be past its best. (The idea, presumably, being that anyone young and vigorous will wish, and manage, to climb the hill; will spend their prime time enjoying the top of it; and then begin to go gently down the far side.)
'Over the top' is the standard phrase here ~ although it still carries a slightly macabre echo from 100 years ago when young soldiers, by their thousands, were ordered 'over the top' of their World War I trenches, never to return alive.

Answer 1 is more appropriate to when something unhelpful HAS (maybe suddenly) arrived, rather than not; Answer 3 is an unusual variant on the phrase 'we're all going to ... '. Answer 4 is presumably from the world of boxing, where 'throwing in the towel' is a ritualised gesture denoting a lack of will to continue fighting.
What with the decline of British manufacturing and engineering over recent decades, Answer 2 has an old-fashioned ring to it ~ but this phrase is still quite widely heard in such circumstances.

Ron may not be a conjuror (= party magician), but this phrase hints at the possibility!
'Up your street' (Answer 2) certainly suggests suitability for the audience, but the point at issue is whether Ron himself quietly has anything ready to do at almost a moment's notice. 'In his cups' suggests he will need to be drunk before he ventures to perform anything (probably not a good idea, unless you wanted the occasion to be memorable for wrong, and probably embarrassing, reasons).
Answer 4 is a 'dummy'; English has no such set phrase.

If someone uses a wheelchair, it suggests they are unable (or uncomfortable) to stand for any length of time, so the pun is on them with Answer 2. A 'standing joke' is any joke which is recognised by 'insiders' within a certain situation or community (e.g. that one person often arrives back late and drunk for afternoon work; or, perhaps, the disabled community does have 'standing jokes' about buildings with poor access ~ but we can be sure they would not use this phrase itself to describe such jokes.).
Likewise with a 'running gag' (a repeated joke, e.g. someone walking across the stage eating a banana at various points during a play) ... the metaphorical sense of 'running' would be uncomfortable for many disabled people.
The remaining two Answers were made-up.

The animal metaphor is the right one here (Answer 2), possibly suggesting either the adding of more cattle to inject more speed, strength and energy into an old farming job, or the mixing of a strong flavour into an otherwise insipid stew.
There was also, meanwhile, of course, the infamous case of British Prime Minister Tony Blair, 10 years or so ago in 2003, who with his aides was seeking to make a dossier more convincing as to the international need for British (and other) military intervention in Iraq. The term they had used became widely quoted as 'sexing-up' the files ~ a stronger and 'saucier' image again.
The other Answers, however agitated they may seem, would not be used in these circumstances.