Before I get into technical stuff - thanks for posting this! I always hope that our products will inspire others to experiment and play with tubes, and the Quickie is especially inviting because it's so cheap ...uhhh... inexpensive, yeah that's it inexpensive. It's fun, and cheaper that taking the family to a movie! OK, now some tech notes:
Max rated plate current (presumably at normal filament voltage of 1.4 volts) is listed as 5mA. Cathode emission goes as the 13th power of filament voltage*, so at 0.7 volts or half the normal filament voltage the max current would be 0.61 microamps (uA). So based on this, 20uA would greatly exceed the current necessary to maintain the protective space charge, exposing the cathode to possible poisoning from ion bombardment. On the other hand, at such a low plate voltage the ion energy is very small, and much less likely to do any damage. Hard to say if that's meaningful. :^)
* This rule is only known for pure tungsten filaments, I've never seen any comparison with thoriated tungsten or oxide cathodes. So it may be quite inaccurate for them!
Assuming the filament is being starved, there should be a saturation current which is the maximum the cool cathode can deliver. You can short the 470K plate resistor with a current meter to see what the maximum is. If it's less than twice the 20uA you are using then the output voltage capability should be adequate. This saturation effect is what causes reduced distortion with starved filaments or heaters - when it's just right you get all third harmonic with no second. As above, at normal voltages this will kill the cathode fairly fast, but all bets are off at these very low voltages.
Grid current begins at a bias voltage that depends on the metallurgy of the cathode and grid as well as the size of the space charge. A few tubes can take up to a half volt or so of positive bias before grid current becomes measurable, while most start at around -0.6 volts. I've measured anywhere from -1.0 volt to -0.2 volts on various 6DJ8s and 6922s, operated at normal heater and plate voltages and currents. Some of the battery tubes are made so that they can operate at zero bias and maintain high grid impedance. For example, the Tung-Sol 3A5 curves show zero plate current at zero bias for grid voltages below about 1.2 volts, which the 12AX7 curves show a tiny current at -0.5v grid bias - all these at normal filament voltage of course.