Running rich when NOT under load

No luck. I know I had air flowing freely throughout the air chamber. There is one very small hole in the throat beside the choke plate where the air mixture needle will protrude when the air mixture screw is fully seated then three even smaller holes in the throat behind the choke plate. All of those are clear and air flowing freely. Air flowing freely front to back on both sides as well. When i first started it, it ran fine for about two minutes, almost long enough, it seemed, for the carb bowl to fill up and then it started running rich again, just like before. It is still running rich at idle, particularly high idle. At high idle the engine will sputter every 3 to 4 seconds and spit out black smoke. It ran fine under load with the PTO. As I left it at high idle it quickly got progressively worse and then when I engaged the PTO it bogged down and died. When I tried to start it again it began to backfire and would not start. I did not continue trying to start it. I removed the flywheel to check the key and it is fine. I'm going to recheck the valve lash. Suggestions?

I just checked the valve lash. I had set them both to .006 but the engine was cold. I set them at TDC on the compression stroke. When I just checked them the intake was at .014 and the exhaust was .017. The engine was warm, not hot. I wouldn't think a cold engine would cause this much of a difference. The set screws were very tight. I did not pull the push rods. I reset the valves to .006. When I cycled the engine by hand the valves appeared to open fully and everything was moving smoothly.

I drained my oil because it smelled a little off and I wanted to check it for fuel. Doesn't seem to be any fuel in the oil but it did seem kind of thin. Of course I'm out of oil so I'll have to go pick some up to see if the valves made any difference.

Fresh oil in. Valves made no difference. It started easy but sputtered bad from the start like it was choked and died within 20 seconds and would not restart. I didn't keep trying, didn't want to wear down the battery for no reason. Suggestions?

Firstly regards main jets.
Back in the flat head days the only difference between the various Hp ratings within a set of bore & strokes was the size of the main jet and the govenor spring
Thus an 12.5Hp had a slightly bigger main jet than a 12 hp which was sightly bigger than the 11.5 Hp which was slightly bigger than the 11 Hp which was bigger than the 10.5 which had a bigger bore than the 10 hp.
I hope that answers the too big a jet problem.

Next , only gasses can burn so the liquid fuel has to be converted to vapour before it can be ignited by the spark.
Some of this happens inside the cylinder and some happens inside the carb.
At small throttle openings most of the fuel will enter the cylinder via the idle circuit as the low speed air passing over the emulsion tube does not generate sufficient vaccuum to suck the fuel up out of the float bowl.
This is why you have an idle circuit and yours is not working as intended.
Idle circuits work in a variety of means depending upon the design of the carb but most of them deliver a rich air fuel mix into the cylinder to be diluted with the remaining air coming through the main carb venturi.
In your case this rich air fuel mix is too rich causing the carb to be running rich under low load = low throttle conditions.

If you ca not chemically clean it fully then you will either have to remove all of the plugs and physically clean it or replace it.
AFAIK the mikuni rebuild kits do not come with blanking off plugs.
I rarely have to order one as I have the ultrasound and if that can't shift the crud, then it is new carb times as the labour charge in drilling out all of the passageways way exceeds the price of replacement.

Thanks Bert, I'm going to find someone with an ultrasonic cleaner or bite the bullet and buy one.

Any chance these symptoms could be the result of a valve sticking or a faulty coil?

Went through it again Bert, I don't know what I could be missing, there is not a hint of blockage anywhere.

A sticky exhaust valve can cause rich running symptoms at low reves.
I have seen them stick when turning over slow but be fine at faster speeds.
Buy an inline neon tube ( red / orange) spark tester and fit it to the engine.
Watch for irregular flashing when the engine is running rough.

So sounds like valves unlikely to be an issue based on the symptoms but can't rule out the coil without testing it. Here are photos of the carb.

That float is not level..

I see the needle seat, have a pic of the needle?

The mower started out sputtering intermittendly at idle, not under load. That got worse and progressed until it got to where it would bog down and die. Then it would backfire when attemting to restart and then would not restart. That seems like a lot to be caused by what is pictured. I have replaced that needle with another good used needle that I had. No change at all and they both held the float at the same level. I can replace that needle with a new one but seems like I may have other problems.