See the linked post for checking the carb inlet, there is a one-way ball valve inside the diaphragm chamber.
It's best to disassemble and clean all the parts of the carb using either clean fuel or carb cleaner.
The air vane fits inside the shaft of the butterfly valve (left side of carb), here is a picture I took earlier. The butterfly valve will be open normally, it closes when the blade part above the flywheel moves up with increasing air flow.
There shouldn't be any need to clean the crankcase with the cylinder removed unless there are bits of the old air filter inside (these would be visible inside the carb & induction housing), I would follow the engine rebuild thread if you have this problem, which is caused by running the engine without removing the old disintegrating foam.
There is a special tool that fits the slot in the top of the cylinder for unscrewing it, you will need to replace the exhaust collector gaskets if you need to remove the cylinder for de-carbonising, they tend to leak if you don't replace them.
See the last post in the engine rebuild thread for a details of the cylinder removal tool.
Taking pictures as you dissemble things will help when you put it back together, I always take more than I need to and not just with engines (vintage electronics is my other hobby).
David