After some reading (quite a bit more than skimming the pages of one of the MIT Rad Lab books -- man, they're dense -- and *useful*!), I suspect the V-I characteristic is roughly exponential, but the toe-in is probably diode-like. There's also an oscillation threshold. And mode hopping (depending on how good the design is), which changes with respect to V, I, B and time -- seeming similar to the misbehavior of laser diodes and other coherent oscillator components (which produce little to no output, or incoherent noise or mode hopping, outside of their ideal range).
V and B are roughly proportional, so that you could get a much lower oscillation threshold and output power level (even going CW) if you had a variable electromagnet, and simply turned it way down. But, er... so, yeah. Pulsed it is.
As for the horn, the dimensions are more or less critical. Its purpose is impedance matching to the air. Its been a while since our RADAR theory courses but I believe speed of RF in a waveguide is 0.9 that of light and it has a lot to do with it.
Yeah, you want to avoid too much SWR. I shouldn't think the dimensions are terribly critical (i.e., accuracy better than 1/10th wavelength -- that's "marking with chalk" to a machine shop) in order to keep it within, say, 1.5:1. More than that and I'd be worried. Don't know what SWR is actually tolerable or desirable (for SWR > 1, the waveguide length will matter, as it will act, at least in part, as a resonator, loading the cavity with some phase shift that will pull the resonant frequency somewhat, or make it hop modes or something), but most anything can handle that, so I can't imagine it'll be *too* bad.
If you can find a TR cell, you can use that to detect RF power, they are usually on the edge of ionisation either from a radioactive gas within, or a voltage applied across an arc gap. Any RF coming through them will cause a breakdown and can be detected. These are used to prevent RF from the magnetron entering the sensitive receiver at TX time.
Beware, the purpose of a TR cell is, of course, to reflect power. So if it's sealed right to the magnetron, you're shorting the output. And there'd be nowhere for the 40kW pulse to go if that's the only load. Which, I think, would result in excessive cathode bombardment.
The whole reason such a small device can handle so much current in the first place -- and also why the heater power is reduced in use -- is because a substantial quantity of electrons are accelerated by the field and thrown back at the cathode, whacking out tons more electrons, greatly enhancing its current density without causing destruction. I would fear that, if the RF were fully contained, it would tend to spark and destroy the cathode surface, resulting in quickly degraded performance and early failure.
Tim