Nice work so far. The screw on the bowl should have a small gasket or o ring on the screw. Double check that. And as far as corrosion on the carb, that can be cleaned with the a good carb break down and scrubbing. As for the jest that’s missing a few threads well of it tightens down you should be good.
Electrical ! Something is missing that when you put the bike in gear it kills the ignition like a neutral safety switch tie in of some sort. Your loosing a ground when you put it in gear. When the break lever is pulled in and the bike is in neutral there is a wire that’s grounded to the starter relay that allows you to start the bike. The same thing is happening for you when you put it in gear that you kill the engine. So start going through the wiring slowly and see if anything stands out. Even a faulty ignition switch so double check that.