Okay, so basically I've conceded that at this point I really don't have the necessary tools to construct a proper DVI switch, which would undoubtedly require the ability to construct and drill printed circuit boards.
Instead, as mentioned, I've decided to press on with the USB switch. I worked on a prototype today, and it seems to work rather well. What is pictured below will switch a single USB device between two computers. It only requires one 4066 because the power will be constantly connected, thus leaving only two data wires to be switched between computers (hella lot easier than 18, huh?).
I had a lot of fun testing this--that is, once I got it working. Turned out that a 6v supply voltage was causing too much of what I guess is termed "on resistance". I tested across closed switch pins and measured ~160 ohms. That was causing trouble for the USB data. I tripled the supply voltage to 18v (still 4 less than the 4066's limit) and that resistance went down to ~50 ohms. It's working like a charm.
What's left is to lay this out a bit neater and then solder it. There will be three of these in total. I will also throw in a 5.1 sound switch and whatever else floats my boat. I can run a network cable (which obviously contains 8 usable wires) to each computer to easily control power, reset, lights, etc. This, I guess you'd call it a VM switch, is still extremely useful given you do have two monitors but would like to use only one mouse and keyboard.
Also, I have some pics of the agonizing process of soldering leads to the DVI connectors. They're so small, and so fragile, it's not even funny.