I've read that GOG has announced that *most* of its games will be compatible with Windows 10 from day 1. How are things looking for the Fallout (1,2 & Tactics) games, which are no longer sold in GOG but are in... pretty much everyone's GOG game libraries? Will they get that kind of support?

Apparently Internet Explorer still exists in Win10 but it's there only for legacy support. The main browser of Win10 is Microsoft Edge (previous code name "Spartan").

There are couple of games that community members have reported having problems with on Win10 listed here. I would not expect any problems with DOS games as they are run thru DOSBox or ScummVM.