I'm going to talk about my real world experiences.
Synthetic oil tends to do better in colder weather. I ride snowmobiles in our cold Alaskan winters and have seen non-synthetic turn to the consistency of raspberry jam.
However, since you are mixing it, the gas will counter that. (as opposed to oil injection)
I've used a wide variety of 2 stroke oils. I would never dream of using a 4 stroke oil in a 2 stroke. It would most likely cause heat issues, plug fouling, and like others have said, smoke. Lots of it.
If you've ever had to clean up 2 stroke oil from a floor and compared it to cleaning up 4 stroke oil, you know that is a much easier to clean up the 4 stroke oil. 2 stroke oil tends to be a bit more sticky.
I've been running synthetic in everything that is 2 stroke, snowmobiles, cycles, weedwhackers, chainsaws, outboard boat and our 2 stroke generator. I usually use the Amsoil Dominator or Interceptor, but have bought the Johnson Synthetic that Walmart sells for $30 a gallon.