If you are talking about Travellerspoint's booking area, then it shouldn't really matter, but if your credit card issuer has stipulations like that in place, then it might. That's something that comes down to the individual credit card though, and I guess you'd have trouble paying with it regardless of website.
You can use someone else's credit card to make the booking, just as long as the booking is in your name. That happens quite a lot (parents/gifts etc.).
The currency shouldn't really have any effect.
Local credit card usually has some limitations, yeah. For International trips u'd better keep another card which accepted all over the world. Yes, other person could pay for you (book a hotel) with their card, usually it's no prob to receive the payment.
I have international VISA which was allright almost everywhere, I only can remember couple of times when bancomat "didnt like" ma visa, but prefered MasterCard. I also use MasterCard -type card (no credit) of one of European countries, and, surpriselly it often doesnt work even inside Europe ( in neioghbour countries)