Heheh, yeah. Certainly wouldn't recommend you do so; but, have seen an amateur build such a transmitter before -- more just to say it's possible, and how well (or badly) it works.
As an oscillator, it depends on cavity dimensions yeah, but also whatever's coupled to the load (so, what's going on in the chamber), and various other conditions. It has some natural bandwidth, of which it mostly oscillates in the middle, but can be perturbed to one side or the other -- this is what gives the tuning range. It's only a 100MHz or so, so the fractional bandwidth is pretty low -- as you'd expect -- but compared to the band it operates in, still a pretty wide range.
Just like, you can pull a crystal, but only by a few kHz; you just can't couple to it strongly enough to push it around further. And indeed, with good reason, else it wouldn't be a good frequency reference, hehe.
I forget why exactly current causes FM. It causes AM too of course; the V(I) curve is not like a tube diode, but a zener diode -- the magnetic field causes a threshold effect, below which the electrons spiral uselessly back to the cathode. So of course, up at that threshold voltage, more current means more electrons blowing the cavities like a whistle, more power output; but it turns out it also modulates frequency. (FM receivers are designed to reject AM, so the AM doesn't matter in practice.)
It would be a bit of doing to put a filter on there, but that is indeed the correct reading, at least as far as I know. You'd need a few more capacitors (at least one more, but probably of a much larger value in order to get useful filtering effect), and another diode, and maybe an active circuit (to regulate the current, saving on filter size). Not recommended, of course; the voltages are nontrivial.

Tim