Couple of tips on your CNS carb.
First off, it has more jets than an NT or Speed carb. It has a main jet, an idle jet and an "enrichment" jet (which is the choke). There is no air choke, the enrichment jet adds more fuel to richen the mix rather than choking off air to richen the mix. I have never had a problem with the enrichment jet, however, if it wasn't closing or was opening up intermittently I could see it causing a problem, so make sure you have slack in that "choke" cable so it doesn't pull the enrichment jet opened.
The Idle Jet has a VERY small opening, if it gets clogged starting will be hard and the motor will run rough at low RPMs. The idle jet is THE thing in the carb that needs cleaning the most, because of the small opening it clogs easiest. I tend to run a very thin wire (like metal wire from a twist tie) through my idle jet when I am cleaning because once upon a time I had used plenty of carb cleaner and the idle jet was still clogged, low end remained rough. Once I reamed the idle jet out starting was fast and low end was nice and smooth.
You apparently have the CNS that has the air mix screw capped off. Though I haven't found my CNS to be particularly sensitive to this adjustment (It's been "set and forget" for me and a ballpark setting seems to be good enough), you CAN pry the cap off of your carb and the adjustment screw IS located below. Drill a hole in the cap and pry it off, then you can adjust the air screw.
The CNS carb should have a rubber seal inside the neck where it mates with the intake. That seal has a special shape that fits over the intake (the intake actually slides into the rubber seal), its not like a regular rubber o-ring. Next time you have the carb off check for the rubber seal in there, go ahead and pull it out and see how it mates with the intake. Make sure you put it back in the carb in the correct orientation. Now the big thing is, when you put the carb back on the intake you need to seat the intake inside this seal properly, otherwise you will get air leaks (and erratic idle). I loosen the mounting screw a bit and pry the neck open enough to slide on the intake without much resistance, then slide the carb on the intake until it bottoms out, THEN wiggle the carb around slightly while pushing it on the intake to make the intake seat that last little bit into the seal. When you are tightening the mounting screw make sure you are pushing the carb firmly onto the intake while you tighten that screw. This should get your carb mounted on the intake without any leaks. I've never needed any sealer or had any leaks installing the carb with this method.
It seems to me like you probably developed an air leak somewhere while you bike was sitting there idling, as I am sure you can see there is plenty of vibration... It could be at the neck of the carb (at the seal I describe above) or it could be where the intake mates with the cylinder, or where the exhaust mates with the cylinder (which would be easier to spot due to residue around the leak and louder exhaust) or you could even have air leaks at the head or the seal on the bottom of the cylinder where it mates with the block, it could even be a seal on the bearing or at the gasket between the two halves of the bottom end. Most of the time the leaks are somewhere in the intake or exhaust, or from a loose head, so make sure your head is torqued down properly to start then check the intake for leaks. I like to use a little high temp gasket sealer on both sides of the intake gasket when I bolt it to the cylinder.
The CNS carb does seem pretty sensitive to the float adjustment, if the fuel level in the bowl starts to dip low the mix may lean out and the idle will head upwards. So make sure you don't have a fuel supply problem from a blockage in the tank, line, filter, petcock or in the needle valve inside the carb...
I know you have checked over all sorts of stuff on your bike, but due to vibrations you probably need to keep checking things all the time cause they will vibrate loose, specially with the stock studs and hardware. I got fed up with having things vibrate loose so I bought upgraded studs and hardware (from sickbikeparts). When I installed the new studs I used red loctite where the studs meet the block/cylinder, and blue loctite on the studs on the end where you thread on the nuts. I also used high temp gasket sealer on all the gaskets before I torqued things down. After doing this my motor was tight a sealed, and nothing rattled loose over the next 1000 miles, to date I have not had to re-torque anything. (which is a small miracle compared to how much I had to re-torque things the previous year) I am very happy with the loctite, that said, some are dead set against it. I will warn that RED loctite is for a permanent bond, and you will need to torch it to remove, so DON'T use red loctite on any of the cheap kit studs cause they WILL strip or sheer eventually, and only use red where you want a permanent bond. That said, I will never run another china 2-stroke motor without sealing it up and using loctite....
Sorry for your frustration, but keep working at it and you will eventually work all this stuff out for yourself. Once you've got it down you'll find it much easier to know what's going on with your motor...