I would start in the carb, as fuel issues (either too rich or too lean) are most likely to cause this collection of symptom. This assumes that you are starting up a cold engine, warming for a brief time then taking off. My first guess would be an issue with the float or needle/seat attached to it. When problems arise here, it can overfill (too rich) or underfill (too lean) the carb bowl. Both can induce conditions that will present to the novice as an inability to smoothly make high power and apparent backfires. It could also be am misbehaving fuel pump. Clearly it is an issue in which heat is changing something and that something is causing the problem. However, in your scenario, most of the parts have not reached that high of a temperature within 10 minutes.
Larry