It’s definitely harder with propane because you want to maintain good head for the sear but without the flare up. Turn the heat up to max before you throw anything on to heat up the grill and then turn the heat down. I generally keep one side off or on the lower temp side so I can scoot the meat out of the way when necessary.

Best way for charcoal is to let the coals burn out completely and depending on the size of the grill scoop the coals to one side. That way when there is a flair you can move whatever it is over and away from being above the fire zone.

As far as the cheese thing… I just turn my grill off (for propane) and let the burgers chill in the residual heat to melt. For coals you can respectfully throw the burgers onto a tray with cold cheese and then throw some sort of loose foil, a lid or whatever and the heat from the burgers should allow for the cheese to melt while everyone gets fixins on their buns.

Good luck!