And if it passes all those tests, first thing when you get it home - check the locknut on the spark advance stop is fully tightened......
If it's a premix 2- stroke and looks like there's an oil slick out the exahaust, turn the conversation to how the seller mixes the oil. I bought an engine once where just putting it in the water would have given the environment agency heart failure because of trhe oil dripping out the prop hub. One twist of the conversation and after confirming he accurately measured out the oil, then "added an extra big gloop to be sure". Half an our on actual 50:1 and it ran much better!
So, with premix an oil slick is not something to worry about. If, however a 2-st oil injection machine has an oil slick I'd be a bit suspicious.
Also take the lid off & check wiring connectors for corrosion etc.