A CCFL inverter could probably work but I don't know if they are current limited to the level that is safe for a nixie tube.
If you tried to light it briefly - like a fraction of a second - if you see a glow it is probably good. There could still be cathode poisoning but that would require longer illumination and looking at each digit in the tube :-/
Can they take a beating, or surpowering is likley to result to explosion?
I think I have read somewhere that a way to get rid of cathode poisoning is to power them over the specifications; that would mean a quick test shouldn't be able to damage the tubes unless incredibly high power is used.
Also, is there a source for english datasheets for IN-14 and IN-19?
I can't manage to put data in google translator, as I don't know how to type cyrilic...
If the tubes are truly NOS then cathode poisoning is unlikely. Biggest risk with NOS is outgassing (neon leaks out through a flaw in the tube). Some people test for outgassing with an old fashioned tool called a sparker. I test mine with an old Radio Shack decorative "plasma globe" lamp. (big glass sphere with gas in it - makes discharges that look like lightning inside). There are all sorts of plasma gadgets that would work. I just turn on the plasma globe and hold the tube close to it without actually touching it. If the tube is okay there will be obvious patches of glow inside of it.
For segment by segment testing I do this: Connect 120VAC through a 0.25A fuse to a bridge rectifier. Put a decent sized electrolytic capacitor with at least 200V rating across DC side of rectifier. This gives you a 169VDC supply with "good enough for testing" stability at a cost of almost nothing.
Hook the DC+ to a 5K resistor in series with a 10K pot and then to the tube's anode pin. DC- obviously goes to the cathode under test. The pot lets you set your current and the 5K is there to limit max current in case you turn the pot to zero ohms.
This is not a super-safe way to do things since it involves dealing with the 120VAC mains with no isolation. So be careful to not hurt yourself!
Edit to add: On your IN-19 symbol tubes Pin 8 is the anode. Voltage = 170. Limit current to 2mA. That'd be more or less 10K anode resistor. More or less because I'm only guessing at the voltage drop across this particular model of tube.
Look into it later when the dust is clearing off the crater.