Yes, they are completely equivalent.

Is 'don't need to' completely equivalent to ' don't have to'?

I agree with the other person, needn't is not widely used here, however, it is not hard to understand either. So, I would encourage you to use it, you inspire others to use it from time to time, lol!

Yes you could answer the question with those three answers, but the first one, 'No, you needn't.' Sounds strange and old-fashioned, at least in American English. I suspect it might be fine in British English.

All 3 are fine, although "needn't" is very uncommon in the U.S., I don't know how it is in other countries.