I'm planning to have the primary LinuxMCE box in my basement, and thin clients behind each tv/display throughout my house.
Yep, that's the way to do it.
Your "thin clients" (Media Directors) are just computers that boot off the core over the network. I believe they are actually "fat clients" but you get the idea.
If you haven't been poking around the wiki, this is a good place to start:http://wiki.linuxmce.org/index.php/Frequently_Asked_Questions
From the Main_Page you can click on Hardware and there's loads of info on what you need.
I will have the TV tuner card in the primary system in the basement, but it won't have any actual display. Would I need a nVidia card for this "basement/master" system or can I just use any old thing for now?
Since the core machine runs kubuntu and needs a compatible video card I would suggest sticking with nVidia. Who knows, you may want to use the core as a hybrid and save on purchasing another Media Director.
1) What "slim" thin clients are there now that I can buy that will attach to the back of a 20" lcd monitor or behind a ~40" 720p plasma/lcd?
I think the new ASUS EEE BOX B202 BLACK, INTEL ATOM 1.6GHZ, 1G DDR2, 80GB machines work for that. There are some initial issues getting the network card (and possibly sound card too?) to work but that can be solved with some manual updates under 7.10RC2. I'm not sure if all that is supported out of the box with the upcoming 8.10 release, but there has been a lot of forum discussion about these boxes with instructions on how to make them work. They are about $375 CDN a box though. I'd love to have 3 or 4 of them too
2) What specs will I need to stream video to 3 remote systems for the "basement/master" system?
Proper networking configuration and infrastructure in your residence. Your core needs 2 nics... the internal network is where you connect all your media directors and NAS devices.
Ideally I'd like it to download torrents automatically to the NAS as well
I don't know if you can make the torrents download to the NAS but you can certainly install torrentflux and have it use a local directory path for downloading, and then copy/move them to your NAS when they are completed (that's what I do).
My advice to you would be to take it slow. Get your core properly set up and running. Make sure your network is wired up right. Then add a Media Director, boot it up over the network, see how that part of things work. Then you can work on home automation (which really isn't that difficult). The phone stuff can be complicated and to me isn't as easy as it looks but with anything, some research and trial and error will get you there. Same with network cameras. Go slow... do things one step at a time. With LMCE I've found you will likely get everything you want to work eventually, but it isn't something you're going to accomplish in a weekend.