My first question is what kind of extension cord (gauge), how long is the cord, and what size breaker is it on. Thinner extension cords and longer length add resistance/voltage drop and can increase amps (or practical purposes decrease the amps available to you). I can pop a 20 AMP breaker if my table saw bogs down on a 2 inch rip cut without an extension cord.
Regarding heat, I would go away from electric. It is going to be inefficient if you go the generator route (fuel to electricity plus waste heat and then to heat). If your power via extension cord is 120 volts and 20 amps that given you a total wattage of 2400 assuming none of the issues I mentioned above. The rule is you should only sustain 80% of that for an extended period of time; a basic space heater is ~1500 watts (my shop uninsulated garage workshop in Maine has a 6,000 watt heater and that only takes the edge off), that is well over half your total capacity and even more for your sustained load. I'd probably go with one of the indoor/construction site propane heaters given your limitations; there are other fuels you could turn to.
For lighting, LED lights draw so little power your budget will not notice them. I have a 10,000 lumen floor work light that draws 100 watts (I cannot currently find the brand online but here is a similar spec'ed item:
https://www.homedepot.com/p/Husky-10000lm-Twin-Head-LED-Work-Light-with-Tripod-HD10000DPT/307939526) For a point of reference in terms of the old incandescent bulbs (google says) a 100w bulb was 1,600 lumens, so 6.25x's the output. There are plenty of smaller clamp on and hanging work lights as well. The brightness and the size are going to be determined by how what you are doing and how far away you placed the lights.
If your power via the extension cord is enough for your needs, either at first or long term, then you can add outlets as simply as a power strip.
For additional power, yes generators are an option. You can get them from relatively small up to power your whole town and in diesel, gasoline, propane, natural gas, and sometimes a combination a couple. If you go that route it will take a little math to figure out what you need based upon what you reasonably will be running at any one time; fuel will be determined by whatever you easily get locally. There are some battery options out there to provide AC power, but again you would have to do the math based on you anticipated needs to see if that makes sense; the batteries could charge when you are not there and discharge when you are.
As for paints and stains, without internal heat (for example when you are away for several days) an insulated garage will eventually freeze. For my shop I built a heated cabinet; it is your basic particle board cabinet that has been lined with rigid foam, then a temperature probe outlet control is installed, that controls a heat source inside the cabinet (in my case old-school a couple 60w incandescent bulbs) which keeps the temperature at 45*F.
Do you already have your tools, do you have a budget for tools? I have mostly corded tools, but they make some amazing battery tools today; I do not just mean drills and drivers, but palm routers and full sized miter/chop saws. They are more expensive than their corded kin, but if they fit your needs they may be less expensive than buying a generator. You could just leave the batteries charging when you are not there ... though you may need a way to charge multiple batteries at once and/or extra batteries to cycle through while you are working.
Basically there are options, they will take time and/or money. Ultimately it depends on what you are going to be doing and your needs and wants.
