Well the round orange or black seals are used with plastic intake manifolds and the cardboard / paper type are used with metal type intakes. Best to get new rubber o rings at local mower shop. If reusing them use a tiny amount of silicone sealer. Just a little.
A bad pto switch can be the problem or a broken wire end inside the pto switch harness. Look closely at each gold color wire end foe a 1/2 missing tab.
A backfire is a pop sound coming out of the carb bore, not really the muffler. What you have going on is either a problem with the muffler coming apart internally or a exhaust valve or seat problem. If you do a good clean carb like you said I would check for these issues.
You need to remove the black plastic lock on mixture screw and remove needle to clean port also. Don't forget to replace rubber orange intake seal at the head along with other gaskets. Or maybe just replace carb with a new carb.
If valves are adjusted right at .004 and stops cranking the compression release built onto camshaft is broken. Needs to be replaced along with case gasket and lower seal. It only takes about 2 hours to do if lower pulley slides off.