Okay, I fiddled with the trem and tried to set it up. At first it didn't really stay in tune well, even in the locked position. But then I noticed that there was much more tension on the D-string compared to the others (to the extent that it was almost impossible to tune it up to pitch with the bridge tuners), so I figured that might have been a problem. Dropped the tuning a step such that there wasn't as much tension and it stayed in tune better after that. Still not perfect, but that may be due to the fact that tension was a little too low on the other strings after dropping them. So I guess it is very important to have the right string tension with this guitar. Then I set the tremolo up with the master tune knob and it stayed in tune much better in this position too now. Also not perfect, but for now I don't want to fiddle too much with it, get the rest of the project started and deal with getting the trem perfectly set up in the end!
I've also put a drone-box on the breadboard today, so now I just need to choose the part values I like best and the I'll be able to solder that fucker up! I may have to order some caps to try out in the noise generator though. I'm basing myself off the CMOS stuff on Beavisaudio:
http://www.beavisaudio.com/Projects/CMOS_Synthesizers/. Still thinking of adding some kind of BMP filter to it though.