Buy your bike before you leave. Whatever tax you might pay in NYC would be more than negated by the difference in the dollar/euro exchange rate. You can get a refund on the VAT you would spend on the bike in Paris, but you have to collect it at the airport before you leave, and the line is verrrry long, (or so it was when I tried to get my refund) so you had better arrive very early if you want it back. After 45 minutes in line, I gave up, and walked away.

Too bad you aren't coming to Japan for your holiday. Most larger bike shops have a large stock of Brompton models. Between a weak yen, and not having to pay consumption tax, you can get a decent deal here. And you don't have to get a tax refund at the airport, just show your passport at the shop and the tax won't be charged.

Flying with a folding bike is no different than flying with an extra suitcase. I advise you to get a case which does not say something like "bicycle transport case", or which has an image of a bicycle on the outside, as many airlines charge extra when you check in a bicycle. When they ask what is in the case, I always say "camera equipment", to avoid the extra fees.