Its only an engine : )
Good on the missing screw, I am sure it is a fixed jet in your carb.
Assuming the float is hollow, did you actually shake it to make sure there was no gas in it? This would still float but raise the fuel level in the carb.
I am not sure and I would expect it to screw up the idle, but if your reed valve leaks, some of the mixture gets blown back into the carb where it picks up more fuel.
If you have been playing with the throttle cable, you may have changed the oil pump adjustment. It is worth checking but unlikely to cause the problem.
I don't remember us ever discussing the choke, what kind do you have and are you sure it is completely off?