Is it normal for the burner in my furnace to turn on and off during a single heating cycle?
Probably. You'd have to inquire of the furnace maker if it is or is not. Speculating, since we have no information about what make or model furnace you have, it probably runs the burner until the heat exchanger is too hot, shuts off the burner until it's too cold, and starts it again. Very few furnaces have any ability to modulate the flame, so it's either on or off. In all likelihood the on/off time varies with the return air temperature (and thus, the heat load.) 

The post-burn run to cool the heat exchanger after the call for heat is satisfied is certainly standard.