I'd rather you did!

Answer 1 leads into the possibility of further discussion without mentioning any specific figures (that's why Answer 2, however convenient, may not be the subtlest approach). Answers 3 and 4 are confrontational and even somewhat offensive ~ unlikely to move any discussion forward 'on the right footing' unless you are very sure of yourself.

Answer 1 is seeing matters more from your own point of view than the friend's, which may be frank but isn't very tactful; Answer 2 is certainly making assumptions; Answer 4 is over-exaggerated.
Answer 3 strikes it about right, and volunteers some safeguards to suggest you have given the idea responsible thought before mentioning it.

Answer 1 is quite a good alternative, but No.2 sounds almost like an accusation, and as such is unlikely to encourage anyone else to help you. Answer 4 also sounds too vague, more as though you were 'talking to yourself'.
Answer 3 is a classic case of English being politely indirect without really disguising what you want. The offer of providing the money (rather than paying back afterwards) is a practical, considerate and trusting idea.

It may seem very odd to you that 'you' should apologise at the outset of the conversation, when the fault is with the other people; but that is 'the English way' ~ possibly rather like people in the Far East not wanting the other person to be embarrassed or 'lose face'.
Answers 1, 3 & 4 may well be 'true' expressions of the situation and your feelings, but they're each rather too confrontational in their various ways. If you are polite, the other person is more likely to be relaxed (rather than defensive) and continue to try and find some scope for agreement to the satisfaction of both of you.

Answer 4 is a bit too blatant about the park being your own choice; Answer 2 presents it as a question for your friend to answer. Answer 1 has rather a pre-planned, assumptive feel about it; Answer 3 seems to be forcing rather too many questions. The best thing is to keep it simple!

Answer 1 suggests a reasonably specific alternative timeframe; none of the others does this at all, or not so clearly.
Answer 2 starts well but then becomes rather dismissive (as though these friendly people would not make the effort to appreciate other commitments within your life).
Answer 3 is very rude and offhanded, and as such would be unlikely to encourage the people to bother with inviting you again; and Answer 4 does not 'leave the door open' from your side to another future suggestion.

Answer 1 is polite, and the 'just a little' allows some space for further discussion; even a slight reduction in the noise would probably be welcome. Answer 2 places quite a lot of blame on the man (not unjustifiably perhaps, but they may choose to take it as hostile personal criticism); Answer 3 makes assumptions that he may well not choose to share; Answer 4 contains a potential threat which he might also respond to in an unhelpfully defensive way.

Answers 1 & 2 are polite, but don't quite strike the necessary tone; a rhetorical question (Answer 3) is sensible but might be taken as sarcastic or provocative. Answer 4 catches the urgency clearly but without being rude.

Answer 2 is the best example of effective English understatement ... particularly if you also offered to pay for the copies, or maybe bought her a small treat to thank her for bothering with it.
Answer 1 seems rather 'high-handed' and old-fashioned in tone, so feels wrong coming from someone in these circumstances; Answer 3 is making persuasive assumptions. Answer 4 somehow seems unbalanced (however sincerely!) with your need for the copies sounding very urgent, but her option to help sounding very minor.

At least with Answer 4 you are postponing the decision (or you hope you are); the others are either blunt or rude or both, especially No.2. If the event is a 'one-off' (such as Bonfire Night) you still risk missing out on a 'slice of British culture'.
Note the several uses of the word 'rather' in these examples!