Check the wires leading to the ignition coils, there's 4 of them behind the passenger seat. Then pull the spark plug wires and make sure you have spark coming from both of them with a test light. Use compressed air to blow out the spark plug tubes to be sure they are free of moisture that might cause and arc type misfire. Next get out the hose, clean the top of the fuel tank (under passenger seat) very well, dry it off 100% and make sure no dirt or debris is present to fall in. Disconnect your battery. Remove the fuel pump harness connector, tank vent line, and fuel pressure line. Use caution, cover pressure line with a rag before removing it to prevent fuel from spraying out. Remove the rubber cover, clean off any dirt you missed before.. Then back the locking ring off counter-clockwise with a driver and hammer, remove the pump assembly and examine the lower fuel pump strainer. If it's brown, it's shot. The fuel filter is the canister which surrounds the fuel pump, if that appears dark inside, it's shot.
Plugged fuel filters can cause all these issues as the fuel delivery is inconsistent at best.
after-fire (explosions coming out the tail-pipe) with black smoke is usually something amiss with the ignition timing allowing unburned air and fuel to leave the cylinder and ignite in the exhaust. Backfiring would be an explosion in the topside of the carb / throttle body caused by ignition when an intake valve is still open. If you have both, they very well could be an actual timing issue but I would rule out the other things first.
My machine did all the things you are describing, so I pulled the fuel pump and found the bottom strainer and fuel filter were plugged. Replaced them and the machine runs now. Just some suggestions, best of luck with your machine!