Woooow. After countless hours of messing with it, I got the sucker to pop up the remote desktop. What I did: in the interface, when the program was reporting that I did have a connection, in spite of the fact that it wasn't showing the remote computer, I right clicked on the remote computer in question, hit advanced options, and changed the monitor from "inherited" to "primary monitor." Whap! Like that, it popped up. No idea why, since it had been working earlier. But if you're running out of ideas, this is something to try. Godspeed!
EDIT: It worked for about three minutes, that is. Connection lost, same old problem back. I'm not a big Teamviewer fan at the moment.