With due dates, I think you should choose what you want. If you're having them at 37 weeks, that is a due date too:-) So, it would be accurate.
With all the questions, people can be so opinionated at times and feel the need to share their feelings with other people. Maybe you can think of short and sweet answers to their questions. Such as the "how are you going to handle taking care of them", you can just say "the best we can" or something simple. If they say "do they have the same dad" - just stare them down until they walk away or tell them that you are not part cat. (I can't believe people ask that!).
Or maybe you can practice changing the subject if they get too personal. You shouldn't have to defend yourself! But some people really make you feel that way. Hopefully other people will have better suggestions.
As for having a big belly, I can sort of relate. I have not had twins, but my 2nd child was almost 11 lbs. I'm small (under 120 lbs at 5 ft 4 in when not prego), and so my belly was HUGE. I measured 43 weeks when I was only 38 weeks. I was literally bigger than some of the mommas pregnant with twins that I knew (which made it worse since I only had one baby). People stared at me like I was about to explode if I went in public. It got old telling people how far along I was and seeing their shocked faces...hehe.
Mostly just letting things roll off of you is the best advice. People will be people and will say thoughtless things (I hope I don't do this to other people!). Don't take it too personal.
Hope the pregnancy goes well!!
PS: So, do the babies have the same dad?