1) In the room with my computer, I have a 46" Samsung HDTV. While I could easily use the monitor screen, I'd prefer to be able to use this TV I just bought a few months ago to watch my TV. How do I integrate it into the system?
Connect the tv to the computer via hdmi, assuming your computer has an hdmi output.
I don't have a network set up, because this is the only computer in the house, so its connection is direct to the modem. Would I need to establish a wired network in order to use my TV with the Hauppauge and run an ethernet cable from the TV to a router?
The tv does not need to be connected to the network. The computer with the tuner does need to be connected to the network, in order for Windows Media Center to be able to set up the guide and download guide info.
Is there any way I could just directly hook the TV into the WinTV without using the TV as my computer monitor?
No, that's the whole point of this tuner, its a usb tuner, there are no video outputs.
Could I take the DVI out and have a splitter with a switch to let the signal go to the TV?
There is no dvi out of the tuner, only usb to a computer.
Would it be better to just build an HTPC and hook that directly into the TV without using my normal computer?
That depends, when watching tv, you wont be able to use the computer for anything else. The advantage to a standalone htpc is it can be used only for tv watching to your tv.