Ok, so you have ruled out fuel/air, spark and timing. Based on the backfire through the carb, it does sound like a timing problem, but you say the key is in place and fine. About the only other thing that would cause those symptoms is valves. Have you checked and/or adjusted your valves? Is this a single or dual cylinder engine and how are the valves operated (chain/belt/gear driven)?
Cheers,
Brad
