I'll give you an analysis of how each one sounds to me (a 20-something Brit)
Will you be having cake?
- Yes, thank you.
Polite way of accepting cake.
- Yes, I will.
Probably acceptable alternative to "1" but I'd expect a little more politeness, like "Yes, please, I will" or "Yes, I will, thanks". Said in the right tone of voice though it wouldn't come across as rude.
- Yes, I will be
Doesn't sound right to me as an answer to the question if you want cake (though I could see it being used), but as noted by JavaLatte it would work fine for the question in the context of a person not offering you cake asking you if you will be eating it (though in the example given of a party I would expect an answer more of "Yes, we will be" since presumably the answerer isn't going to be eating all of it!).
- Yes, I will do.
Sounds slightly off when analysing it but I probably wouldn't bat an eyelid if I heard it as a response to the "do you want cake" question.
Will you be late?
- Sorry, I will.
Seems fine to me as a polite answer if the organiser or someone who would otherwise be disappointed by your absence is asking the question. Other alternatives for polite answers would be "I'm afraid I will".
- Yes, I will
- Yes, I will be
- I will.
These I would expect more in a case of when someone not in a position to complain about you being late asks you the question. Like a friend asking "Will you be late for that meeting you have with your boss?". These could also be slightly less polite/slightly more terse answers, though again, said in the right tone of voice, they wouldn't come across as positively rude.
Are any of these short positive answers "wrong"?
Here is the list of short negative answers.
Will you be having cake?
- No, thank you.
Polite answer.
- No, I will not.
This one sounds rude/abrupt to me. "I will not" tends to be quite emphatic these days. "No, I won't" is perfectly fine to my ears and less insistent.
- No, I won't be
Again, not impolite, but not particularly polite either. Said in the right tone of voice you wouldn't take offence to it, and again, it'd be used in situations when you aren't being offered cake but instead asked out of curiosity.
- I won't, thank you.
Equivalent to 1 to my ears.
Will you be late?
- No, I will not
See 2 in the cake question; this would come across as emphatically denying your lack of punctuality, and could be interpreted as you being defensive/getting cross at the suggestion that you would be late.
- No, I won't
- I won't be
These two sound equivalent to me and would be the answer to the question used in most cases.
- Are all the answers idiomatic? Are any non-standard?
The only one that sounds slightly off to me is "Yes, I will do" but as I said I probably wouldn't even notice if someone said it to me.
- Are there any other acceptable alternatives?
Plenty besides the examples I've listed above. There are all sorts of embellishments you can use. In a lot of cases a simple "Yes, please" or even just "Yes" would suffice (though again, depending on tone of voice the latter could come across as being terse and a little impolite). To express reluctance and the idea that you shouldn't really be eating cake but you want to anyway (or even to mock this idea!) you could say "Oh, go on then!"