A regular gas engine is not designed for avgas. If you have a cat it'll be toast immediately, if the avgas is 100LL, since Low Lead is actually not so low on lead (my more than old leaded fuel). Also, long term use can leave deposits on the exhaust valves making them stick in the guides. However, if you top your normal gas with some 100LL/avgas you needn't worry.
On a side note I've run my lawn mowers on avgas, Briggs&Stratton and Honda 4-strokes, for years with zero issues but maybe they are looser internally. Why? Because the slugs hate the smell/taste of avgas exhaust fumes so they're all at my neighbours place now...

As for your engine/tranny build I suggest you take it slowly. If you continue throwing parts at it chances are you'll end up with a combo that simply won't work.