I hold a Schengen visa, which is valid for 3 years. Can I work in Georgia on this Schengen visa without having any Georgian visa? If yes, for how long can I work in Georgia on a Schengen Visa?
The best way to answer this question would be to work through the official "Do I need a visa?" wizard.
The short answer seems to be "Yes", on the basis of Ordinance 256, which gives a list of countries "whose visa and/or residence permit holders may enter Georgia without a visa for an appropriate period and under appropriate conditions" "for 90 calendar days in any 180-day period."
To address the obvious point, I worked through the wizard on the basis of being a Moroccan citizen* with a French visa visiting for work purposes and it gave me a green light and a link to the text of the ordinance, so it seems that this ordinance does apply to working and not just to tourism.
However, it's still a good idea to work through the wizard yourself in case Georgia has some particular dispute with your country of citizenship and has made an exception in a different piece of legislation.
* I just picked the first country I thought of which wasn't in the list.