Carb blows black smoke and runs bad but run good with no air filter or with a new filter

Have you tried removing the air filter element? If this does not help, take it to a PROFESSIONAL mechanic and have him clean, set, and synchronize the carbs!!

Is your choke stuck on? The float in your carb could be stuck.
Black smoke means too much gas being burnt.
Make sure your air filter is clean. I saw a car that would not run because the air filter was plugged.

Flooded.... Take the plug out and let it air out for a day or two OR if you have an air compressor, take out the plug and blow air into the spark plug hole to dry it out. Gas+spark+air = should run. but if you have too much gas, it won't.
Good Luck.

Can be a bad O2 sensor. Check with a scanner which shows live data the 1-1 and 2-1 (upstream) sensors. Good sensors show a rapidly fluctuating voltage from 0.2 to 0.8 V. When the sensor has a steady voltage, then its most probably bad. There is no check engine light and no fault codes set?

white smoke would be a sign of a head gasket blue smoke is a sign of burning oil black smoke is a sign of a fuel problem so blowing smoke and rough running do a compression test and look for a weak cylinder

This sounds like a micro switch acting up, the first switch to check is the seat switch, make sure the block connecter is pushed fully home on the switch ( there is often a fail safe inside the connecter should the block come away from the switch )

Black smoke indicates too rich, still. Verify clean and dry Air filter.

Looks like you've got a good idea on what makes diesels smoke, so I'm sure you know that black smoke is almost always from too much fuel for the air that's going in. So it can be too much fuel or too little air. If your filters are good, the next things to check are the fuel pressure and the injectors.

I know diesels, but I don't have any specific experience with lawn mowers (kubota?). Since it stalls eventually, it sounds like it's not a single cylinder (although not for sure) so I'd check injector pump first.

take off the bowl of carb....the main jet is clogged ....remove it with a screwdriver...a small piece of wire will work as something to clean it out.....clean the whole carb....fixed many of them...

Make sure it's not overfilled with oil.
if this is not the case, then you have excessive blowby, which is what blows past your rings when the air/fuel mixture ignites. there is only one explanation for this, faulty rings, and that's what it sounds like to me. the blowby forces the oil into the vent leading to the filter, and back into the cylinder, where it burns, creating blue smoke.