If I can't narrow down any of those variables via research, I'll attempt to build a vortex gun that can change as many of those as possible.
Here's my plan:

Gas pressure in chamber: set at air-compressor/compressed-gas-cylinder via gauge/regulator

Chamber volume: I plan to copy shiddymunkie’s ...

I'm working on a pneumatic vortex ring generator too - of a larger scale, like shiddymunkie wants for his next ;-)
Unfortunately, I’ve found limited information on how vortex rings actually form.
Ring variables that I can think of:
1. stability / longevity before it dissipates
2. forward ...