1. Yes. Anything supporting Gigabit placed on the Gigabit switch will run at Gigabit speed between other Gigabit devices. But of course it won’t help for any traffic that leaves the Gigabit switch and has to traverse the 10/100mbps switch (if that’s part of your plan).
3. Cat5e is fine, although at this point I personally would invest in CAT6 if I was headed down the Gigabit path.
4. Assuming you keep the Gigabit devices on the Gigabit switch, and 10/100mbps devices on the router’s switch, probably not. For streaming content (even HD), 100mbps over wire is plenty, even for 2-3 concurrent streams, although I assume you’ll likely only have one at any given time. And anything downloaded/streamed from say Netflix or Amazon is not only going to be limited by the 10/100mbps switch, but the Internet itself (iirc, about the most Netflix reaches is ~4mbps). So the Gigabit switch is really only going to help file transfers between the NAS and HTPC, but I have no idea how often that occurs, if at all (not counting streaming). And even if it did, it’s confined to the Gigabit switch, so who cares.