Clearly Rectifier has done his research and has learned through direct experimentation. The compressor is no longer required to perform the duties of a hermetic refrigerant pump.
When repurposed as an open loop air pump (air compressor) the lubrication requirements are exactly as he stated: "non-foaming, non-conductive, and light enough to be sucked up by the hollow crankshaft". Additionally, Rectifier mentioned that using a compressor indoors can be problematic due to oil vapors. Mineral oil is the most logical choice.
Rectifier, I wish I could tell you the oil capacity of your compressor but you probably already know that having a manufacturer's name, part number, and the Google search engine will get you nothing but Chinese export salesmen trying to sell you a pallet of compressors.
You probably already know that the vacuum ports are not isolated from the oil supply like the pressure port. Too much oil will block off the vacuum ports or cause leakage. You could clean out the old oil and determine the maximum capacity of your compressor in one operation: Fill a graduated container with a suitable solvent. Cut an opening in the unused pipe stub (the #3 factory crimped pipe). This is referred to by several names but it is essentially a vacuum port and it is at the same level as the original vacuum port. Hook a funnel and some tubing into one of the vacuum pipes and begin pouring solvent into the funnel. Stop pouring when the other vacuum port begins bleeding solvent. Record the result and now you at least know the absolute max capaity of your compressor and can avoid overservicing it. Reseal that third pipe and affix a non-collapsable hose to your #1 vacuum line. Fill a container with your calculated amount of mineral oil and submerge the vacuum hose in that fluid. Turn on the compressor (yes, there is no oil in the sump). Watch the cup of mineral oil; an airtight and healthy compressor will suck up all that oil in 1-2 seconds. Of course, shut off the compressor if it fails to scavenge the oil. Good Luck sir and keep us informed!

It depends on the compressor: An R12 type system uses a special form of mineral oil, while an R13 system uses either an Ester-based oil or a PAG oil. Oils not made for the compressor could break down after continued use.

the lowest friction one..... that would be the obvious choice

I don't know if a fridge compressor is different but we just chucked light engine oil in our big compressor from time to time, even hydraulic in a pinch and it's thirty days older than me, well the pump is, the engine was swapped when the exhaust detonated due to age, then fell off on the way round so the new silencer was a bit of pipe welded directly to the manifold... My guess would be that two stroke wouldn't murder it but it may be a little heavy, I don't know much about fridge compressors as I said, however it may need a few repairs if the oil's coming out like that, it should be trapped inside it. Assuming it's piston based that is...

I'd think ATF, power steering fluid, or hydraulic fluid.

I couldn't help you with your oil problem, but I sure learned a lot in the process of discovering that I couldn't help! Good luck with your project.

. Part of the reason that a special oil is required is that there is a partial vacuum in the compressor and one needs a lubricant that won't lose the more volatile components. It also requires a lube that won't be affected by the refrigerant. . I'd stick with lube designed and labeled for A/C compressor use. You can usually find it at automotive, hardware, and larger department stores. It often comes in the same size/shape ~1# can as refrigerant, so you can use the same equipment.