Re: Honda BF20 2005 four stroke rough running & stalling.
About all you can do with those screws after they're "boogered" up is to use a Dremel with a cut off wheel and make a new slot for a flat blade screwdriver. They will usually come then because the heat generated from the wheel cutting the slot expands the metal and then it shrinks a bit when it cools. If not, then use the wheel to completely remove the head and, with the bowl off, you can grab the nib of whats left of the screw with Vise-Grips or pliers. WD-40 and Liquid Wrench are not very good for anything but cleaning. Use a product called PB-Blaster for penetrating purposes. Available at most auto and hardware stores where you're going anyway to buy a new screw. Works like magic on most things but not on those screws.
One thing I didn't mention in my "detailed" response is that you should use that carb cleaner straw and spray into the hole where the "jet set", with it's oring, pushes up into the top of the carb throat. That is the main passage feed for all the circuits except enrichment. If yours is electric, the enrichment is from the carb bowl through the electric solenoid pin. If it's a manual choke, there is no enrichment circuit to worry about. I'm guessing yours is electric because of your symptoms. Is that right?
You should, once again, prop open the throttle plate so that you can observe the liquid coming from the three tiny orifices at the top of the throttle bore. The one that you can see when the throttle plate is seated or closed is idle and the the other two are the high speed and transition holes. All three should have small 1/8" to 1/4" "gushers" of liquid coming out of them when you get that little plastic tube seated and are shooting the juice to that passage. The high and intermediate are easily "backflushed" and cleared but sometimes you will need to remove the idle mix screw completely (real pain in the butt) to get it to flow properly. It too uses the little "ITEM 1" oring and that should be replaced if disturbed. If it were mine, I'd reseal the whole thing with new gaskets.
Let us know how you make out.