No, Thank You - Polite Negative Expressions

You should not be too brief (Answer 1), and you are not obliged to give your reasons (though the other person may well ask you exactly that, as the next thing they say; so be prepared!).
Answer 4, clearly, is just plain rude.

You need to express both the impossibility, and your regret. Answer 4 does this most clearly. Answer 2 is blunt and rude; Answer 3 suggests your host(ess) is at fault for not having realised the problem beforehand.
You needn't specify who has forbidden this to you: it may be obvious, or perhaps a private medical matter that you are not obliged to disclose in detail in a social context, such as round the table.

It is confusing if you reply with a 'yes' (Answer 1), even if you are affirming that they are right (i.e. that you didn't want any).
Answer 2 is clear, but rather blunt and therefore sounds somewhat rude. Much of Answer 3 is 'filling', but it makes your refusal softer and more socially acceptable.
Answer 4 is far too strong to be polite.

Answer 1 does not actually offer a clear reply to the Question; would you like to go, or not?
Answer 2 may well be true, but nobody will feel particularly comfortable if you say so.
Answer 3 is a perfectly reasonable delaying tactic ~ 'buying some time' ~ but clearly they will need a firmer answer before long, in order to confirm the logistical arrangements (whether or not you will be joining them).
Answer 4 remains the clearest, even if it does not explain any reason. That's all they asked you, in the first place; and if your answer is negative, why should the reasons for it matter particularly?

Answer 1 may happen to be true (and fairly final, as far as your own attitude to the animal is concerned), but it is hardly a sensitive reply that offers any common ground. Answer 4 is rude because it is dismissive and even confrontational.
For the purposes of this Quiz, we 'aren't saying Yes', so Answer 2 is irrelevant.
Answer 3 turns down the offer, but hints about a more friendly answer in the future (even if both people are fairly sure that it isn't seriously true).

Answer 1 is clear and negative; Answer 2 is also clear, but if the man means you no harm, he might be offended by it (even though he ought to understand your reasons). Answer 3 may be unnecessarily harsh; Answer 4 may seem polite, but is quite obviously dishonest (raising false hopes for him).
Obviously, if the situation were to 'turn ugly' you would probably say something else which we might prefer not to put here, and/or begin shouting, and/or run away as fast as you can.

This is probably the clearest and politest. You have made it clear that there is nothing wrong with what she is offering you, so she should not feel in any way to blame.

You would probably like to say something similar to Answer 4, but Answer 3 makes your position clear without hurting their feelings.

Answer 1 is the clearest; Answer 2 may be true but is blunt and rude (this person is only trying to do their job, 'in a good cause', after all); Answer 3 may happen to be true but is rather a 'thin' excuse. Answer 4 is a kind thought, but neither we nor the charity collector would expect you to say so much.

Answer 1 may seem slightly silly in a serious situation, but it may be better than turning confrontational with a person like this: the situation could quickly become ugly. If he or she does not take the hint, you may then need to say (or do) something stronger.
Answer 2 (or something like it; ' ... or were you only joking, anyway?') might be polite and clear but not too confrontational.
Answer 3 is probably too much to remember clearly in a sudden and worrying situation, but it's the sort of thing a self-confident native speaker might use.
Answer 4 is good and clear, but it may be worth a 'softer' deflection before you say anything as harsh as this yourself.
We certainly do not wish to suggest, in this Quiz, that you are any more likely to find yourself in a difficult situation in Britain than elsewhere in the world ~ but we hope this has opened your eyes to some more of the ways of saying 'No', with various levels of clarity, politeness and subtlety.