"BTW, I'm messing around with finger positions (based on Carolina Eyck's book and videos), and it seems like I have to have an extremely narrow field in order to span an octave without moving my arm. Is that me doing something wrong, or just the nature of the beast?" - Asbjorn
It's the nature of the beast. If you have an analog Theremin and you want a linear pitch field, you usually end up with ~1 octave open/closed hand. Whatever you do, don't breathe! :-)
I've got my (digital) Theremin set to 1/3 octave open/closed hand, which makes vibrato easier, and I don't have to stand stock still like a statue. But I suppose it gets in the way of / precludes the use of the various aerial fingering techniques developed for higher sensitivities. I keep my right thumb and index finger tips loosely together, with my other fingers out, in a relaxed "OK" (or, in these days of unabashed Nazis roaming the earth, "White Power") configuration. I "flutter" my entire hand at the wrist for vibrato, and sometimes hinge my loose fingers all together at the main knuckles for quick note stepping. I'm nowhere near as exact as many players, but have only been playing for ~1/2 year, and the pitch display and pitch correction make up for a lot of it. And the pitch field on my Theremin is ruler flat, even at the antenna, which also really helps my playing.
If you pick slow pieces everything gets a lot easier! And IMO you really need some kind of instantaneous pitch feedback, whether it be my visual display, pitch preview earphone, or speaker right behind your head.