the thing where the symptoms are more extreme after a longer period of sitting is weird. BUT, to me what you are describing when the motor is cold sounds like the carb is tuned pretty lean and while it's cold it is really lean. that's why it's backfiring through the carb until it warms up.
idle mix and transition circuits are probably leaner than the mains and thats why it clears up after 2000 rpm while cold.

is the choke on the carb operating?

fyi my carbs are tuned lean enough that the motor acts grumpy like this when it's cold. since it's starting to get colder here i'm probably going to richen up the idle/transition circuit a step so that it's a little more driveable in the colder weather...