If it is backfiring only when you kill the engine, allow the engine to idle a few minutes before you kill it. If it is backfiring while in use, check the intake gasket and bolts for wear or looseness.
The best answer for this is: Get a tune-up.
If it is backfiring the timing is set incorrectly or the spark plug wires are not connected in the correct order.
Rev the engine six or seven times before coming to a complete stop. Rev it all the way to the edge every time you stop.
well if its a 2 stroke it needs new reeds if not, your timing is prolly off a bit
More than likely you have a exhaust leak.Check the headpipe seal
Backfiring it normally caused by a car that is very badly out of time. It can also be caused by the spark plug wires being connected incorrectly. Switching to a higher octane will not help at all.
Flywheel key (or shear pin) can shear off which will put the motor out of time and wont start . It could do damage to some other parts but that is the most common.
Check the air filter, fuel filter. Either of these easily blocks on mower.
Most common, a rich fuel mixture will cause the backfiring. Check the condition of the air filter or close the needle valve on the carburetor 1/8 turn. Also, a vacuum leak will cause backfiring. Check the intake manifold for loose bolts or worn gasket.
NO, but a good tuneup will probably cure your problems. There is no cure in a can for the symptoms you list.
mower manuls are at this link http://manuals.mtdproducts.com/mtd/DocGetter?doc=769-00533.pdf