Backfiring, rough idel

The alternator is used to charge the battery. There is no reason to believe the replacement has caused the fuel utilization problem, or smoke from the tail pipe, or rough running engine. A good mechanic should be able to examine your engine and find the problem. White smoke could be valves or piston rings worn out. There are many things that cause rough running engine. Fuel mixture controls, fuel leak, faulty timing, there is a long list. Good luck.

Color of the smoke? White, white with a blue tint, or black.

White smoke is water vapor and in this cold climate it is normal. Strong gas smell is not normal although you may smell some while engine is warming up. Bad cat converter can produce smells, but usually more like rotten eggs. If engine is running rough and seems rich, check coolant temp sensor. Bad converter can cause stalling too. Otherwise, idle speed is controlled by idle air contol motor which can get dirty and stick. Any codes set?

White exhaust smoke from a gasoline engine usually means coolant leaking into the engine block.
Is the coolant in the radiator full ?

Faulty exhuast cam actuator control solenoid.

check for water in the radiator at the cap
indicates that it is over heating

Hi,
It sounds like you have alot of problems which may only be a couple.
Have the coolant thermistor checked, it sounds like this is not switching off and causing the engine to over fuel, making it run rough and blow black smoke out the exhaust.
Have the engine thermostat checked it maybe faulty causing the engine to over heat., have these done and see how you cars goes then.

Should still be under warranty. Take it back to the dealer.

if u have a damaged plug it can be running on 1 less cylinder ,it will idle very rough,change it!

If smoke is sweet smelling and alot of moisture drips out of tail pipe, it is probably engine coolant.Check for coolant loss.