Briggs 26 HP 3 problems

It is on a Bad Boy ZTM, 2007 year but only 442 hrs. Has set for a few years.
Replaced the old carburetor.
Problem 1:
Will only start if I squirt a little gas in the air intake. Then it starts fine and runs good and will restart..Choke? Fuel pump or hoses ?

Problem 2:
After the motor gets hot it begins to backfire and run rough. I don't know if the plugs have ever been changed. Coil?
What type plug is best? The guy that serviced it in the past is out of business. If I add gas it stops backfiring for a while (maybe it just cooled down a little?)

Problem 3:
A mouse built a nest under the engine cover. I took it off and cleaned it as best I could, but it smells very hot after running a while. So, can I clean this engine with engine gunk cleaner spray, then wash it off with a hose, or is that a no no?

1. Is the choke working on the new carburetor. I have seen the 'wrong' replacement carburetor installed and the lever on the choke itself was missing the hole for the link from the throttle quadrant. (hope you still have the old carburetor to compare. basically some carburetors are set up for one type of choke operation only, and others can work the choke both ways, off the throttle lever or a separate choke cable.

If your choke works off the throttle lever I have also seen the throttle cable out of adjustment and not pulling the linkage on the engine enough to push the choke completely closed. If this is the way yours is set up, look for a lever about 3 long opposite the throttle cable. push the throttle to choke position, then with your finger, see if you can push that lever up a bit more. If you can push it upward, pull the throttle off 'choke' and adjust the cable at the clamp so it can pull the throttle quadrant around a bit more.

2, 3.: These are probably related. And they may be related to your mouse problem. Since you say it is worse when hot, try running with first one and then the other spark plug wire off the plug. If you have an in-line spark tester, you can also put that in the circuit and see if the light blinks more on one side than the other. (At wide open throttle there should be very little if any flickering. Much flickering at all indicates that coil is not making a spark consistently.)

It would also probably be a good idea the check your valve adjustment. Assuming yours is not a Vanguard engine, the adjustment should be .004-.006. (I usually use .004) with the piston about 3/8 down the cylinder on the combustion stroke.

If you isolate one or the other cylinder as the problem and you find you still have a good spark, you may have a problem with a valve guide or seat loosening up when they become hot. (this happens a lot when mouse nests cause a cylinder to overheat. Once it starts loosening when hot it is almost impossible to guarantee it won't happen again soon without replacing the head.)

One other thing that occurrs to me. Did you see any bare wires between the ignition coils when you fixed the mouse problem? Mice sometimes chew the insulation off the wires and then they can short circuit the ignition coil or coils.

That kill switch wire has diodes in it. Those diodes keep the circuit in one coil from killing the spark in the other coil. If you determine that your engine is losing spark, another test would be to disconnect the kill switch wire, put the cover back on and run the engine. I am thinking your engine has a fuel solenoid, so you can still stop the engine, it will just run 5-10 seconds after you turn off the switch. (Basically it uses up the fuel above the fuel solenoid)

We once had a mouse nest under the flywheel, had to pull it and clean it out. I would not use water on the engine. Like TRK said, check the valve clearance, it probably has a decompression feature so follow the instructions carefully.

The hard starting sounds like the choke is not working correctly.

Mice are your other problem. if they pee on the coils the coils are ruined.

The no start and back firing when hot will most likely be traced to a valve clearance problem.

When you open it up to check the lash. Go ahead and pull the push rods and check for bent ones. Pay special attention to the intake rods. For some odd reason B&S used aluminum push rods on the intake and steel ones on the exhaust. If you find one bent. Replace it/them with the steel exhaust push rods. Both are the same diameter and length.

Thanks to all four of you for the comments.
The starting problem was the choke adjustment. Moved the cable, oiled it, worked at many times, freed it up, engine started great. Thanks
next problem; I don't think the throttle is opening fully probably need to adjust the cable, will do that tomorrow.
Put the inline spark plug tool in place. Steady light until the engine gets hot and begins to sputter, then the light flickers a little. Tomorrow I plan to try to clean out the engine fins with wire brush tools.
How do I determine if it's the coils.
You guys are way over my head when it comes to adjusting valves. I'm hoping I can get this done with just the coil replacement.
Thanks so much for your help.

I just completed fixing a similar problem. A rat decided to stay warm under the top cover, on top of one of the cylinders and obviously blocking the air flow to that cylinder. It must have been there for a long time without me knowing it because when I started getting intermittent missing, especially when the engine warmed up, and removed the shroud in troubleshooting to find the rat, it was crisp and well decayed.

In short, after going through a lot of troubleshooting and time, it turned out to be the ignition coil on that side. With what you described as lack of normal maintenance, you can add potential problems to your list from that....servicing is where I would start in troubleshooting.

Also pull the plugs and check for even burning on both. If one has significantly more carbon buildup, if the problem is your coil, its the coil on that side.