Out of curiosity, is this the style of carb you have, and if yes is the screw circled in black the idle screw you are adjusting? I ask because generally if the idle mixture screw is located behind the throttle slide, it is a fuel screw and not an air screw. The more you turn it out, the more fuel you get. This may be why you are having so many issues.
It is also worth keeping in mind that, while the pilot jet size does affect the idle mixture fuel volume, you can have a 2 or 3 pilot jet sizes that are close to the right size that will allow an idle mixture screw setting to be reasonable and idle, but not run correctly at just off idle to 1/8th throttle. It's important to distinguish the two because a slightly too lean pilot, and a slightly too rich pilot, can both have idle mixture settings that fall within the reasonable range of adjustment. A good example of this was on one of my bikes with a Mikuni carb. A 25 pilot jet idled perfect with the idle mixture screw (fuel) turned out 2 turns, but at light steady throttle it would hesitate and stumble from being lean. A 30 pilot jet idled perfect at 1 1/4 turns out on the idle mixture screw, but was rich enough to turn the spark plug black. I had to run a 27.5 pilot jet to get a decent part throttle fueling and a good idle mixture.
Tuning rule of thumb with fuel screws. Less than 1 turn out from seated you need to drop down a jet size. More than 2 turns out from seated, up a size. Don't ignore those "half" sizes. like in my example above where the 27.5 jet worked the best - in Keihin terms, the jets that end in 2 or 8 are the "half" sizes between the 5's and 0's.
It is also worth keeping in mind that, while the pilot jet size does affect the idle mixture fuel volume, you can have a 2 or 3 pilot jet sizes that are close to the right size that will allow an idle mixture screw setting to be reasonable and idle, but not run correctly at just off idle to 1/8th throttle. It's important to distinguish the two because a slightly too lean pilot, and a slightly too rich pilot, can both have idle mixture settings that fall within the reasonable range of adjustment. A good example of this was on one of my bikes with a Mikuni carb. A 25 pilot jet idled perfect with the idle mixture screw (fuel) turned out 2 turns, but at light steady throttle it would hesitate and stumble from being lean. A 30 pilot jet idled perfect at 1 1/4 turns out on the idle mixture screw, but was rich enough to turn the spark plug black. I had to run a 27.5 pilot jet to get a decent part throttle fueling and a good idle mixture.
Tuning rule of thumb with fuel screws. Less than 1 turn out from seated you need to drop down a jet size. More than 2 turns out from seated, up a size. Don't ignore those "half" sizes. like in my example above where the 27.5 jet worked the best - in Keihin terms, the jets that end in 2 or 8 are the "half" sizes between the 5's and 0's.