* I replaced the pto switch, because I knew it was going bad. That didn't help.

* The battery is new, and it went dead after it was sitting in the garage for a while. I'm not sure if I left something on, but it seemed wierd that it went dead. So I thought there might be a short somewhere. I charged it back up, and it's been good since.

* There's plenty of new gas in the tank. It could be mixed with some old gas, but I don't think that's a problem.

* After I've been turning it over for a while, I smell gas. So I think it's getting fuel, and maybe flooding after a while. But I don't know how to tell for sure.

* I checked for spark, and I think it's ok. But I'm no expert at these things.

* I squirted some carb cleaner in the cylinder, and put the plug back in. I eventually got a sound that sort of sounded like it was trying to fire. But it only happened once, and it wasn't really loud.

* I put my thumb over the spark plug hole, and cranked the engine. There was very little pressure. But I read that the Kohler engine has ACR (automatic compression release), so maybe that's normal?

* I took the plastic cover off the engine, and found a large nest of mice. Unfortunately, I think they got toasted. And there were 2 wires with the insulation removed. I don't think the wires were broken, and I'm not sure if they were chewed. They looked more like they were scraped. They're connected to this thing that looks like it has magnets on it, that's next to the flywheel. I think it's a magneto--whatever that is--but I'm not sure.

* I squirted some carb cleaner in the carb, but no luck yet.

Does anyone know what could be wrong?

If the engine still remains in the non-running state, give the carb a shot of carb cleaner (as you did before) to rule out lack of fuel as a second (unlikely) cause of non-run.

If the engine still does not run after this, remove the spark plugs and hook the plug cable back to the plug. Lay the plug on a metal part of the engine (not the flywheel) where you can observe the spark gap.
Crank the engine and watch for spark. If you see a good spark (you might even hear it snap), alas, the engine has probably sustained some serious damage from the overheating.

As you mentioned, this engine DOES have ACR, so a standard compression test is not recommended for determining the condition of the cylinder combustion chambers.

A cylinder leakage test (described in the Kohler manual), using compressed air can pinpoint where the loss of compression is occurring. But a cylinder leakage test requires that the individual cylinders be positioned at a specific spot in the 4 stroke cycle or the test results will be misleading or downright useless.

Complete the tests of the ignition system and post back your results.

So, I disconnected the kill wire for the magneto thing--that wire was connected to the chewed stuff--and it didn't make any difference. Then I reconnected the kill wire, and squirted some carb cleaner in the carb, and tried again. After it turned over a bunch of times, I got one really loud pop. After my ears stopped ringing, I took out the spark plug. Put my thumb over the hole, and turned the flywheel by hand in both directions. I didn't feel much, if any, air pressure. But I heard something that sounded like air being released. That may have been normal operation of the ACR though. I didn't try checking the spark again, since I was a little concerned after the explosion from the carburetor. I figured I'd wait a while, and let all that stuff dissipate first. But I'm pretty sure, since I checked it before, that there's enough spark to cause the engine to fire at least a little. And it could be a normal spark as far as I can tell.

Ignition system.
You have stated a couple of times that you believe the ignition system is producing spark. The bang that nearly deafened you (soiled BVDs?) proves that you at least had one instance of spark. So, we might want to presume (for the moment) that the ignition system is OK.
Disconnecting the kill wire from the magneto eliminated the possibility that chewed or bare wires were stopping the production of spark. Place a question mark or 2 beside "Ignition system" on your list.

You have indicated that you feel little or no compression when you place a thumb over the spark plug hole and turn the engine by hand.
Do this. (you might need a helper) Move the spark plug wire as far away from the spark plug hole and place your thumb over the spark plug hole. Spin the engine using the starter.
There should be enough compression that it takes a bit of effort to keep your thumb held over the hole. If you do not feel the compression trying to push your thumb off the hole.........the engine is not making or holding enough compression to run.

If this is what you find, it is time to decide if you want to go further, or whether you would even be able to go further depending on how well equipped you are to perform a cylinder leakage test.
Read and study the Kohler service manual page 3.4 about the cylinder leakdown test.
Though the Kohler leakdown tester kit is nice, you really don't need that much capability.
But you will need a source of compressed air, an adapter fitting for the spark plug hole to mate the compressed air hose to the engine cylinder head, and you must have socket wrenches large enough to fit the flywheel retention nut so the engine cannot move when the compressed air is applied to the engine.
If you qualify in regard to the above, and feel confident/mechanically competent enough to do a cylinder leakage test, then you can do one to see why the engine has insufficient compression.
E-mail me for a copy of my description of how to do a "poor boy's leakdown test". The poor boy's test still requires the above tools and compressed air, so if you do not have (or can borrow) those, you do not qualify for a DIY leakdown test. Place a "frowny face" next to "Compression" on your list and choose a professional small engine shop for further testing.

I just bought an old mower from craigslist, and I think it'll get me through the rest of the season. And I needed one anyway, so I could collect grass and weed clippings for my garden. I read through the tractor manual, and it sounds like the problem could be something like a stuck valve. So I'll probably try some kind of oil additive stuff, that might be able to unstick things. I figure it's worth a try.