Sounds like its running too rich with the backfiring, the hanging idle could be part of that as well. How did the plug look? Was it black and wet or? Reading the plug is a surefire way to know whats going on in a motor.
I would say you still have an issue in the carb, did you pull the jets and really go through it, blowing compressed air through all the passages and soaking it good? I have had to let them sit in a can of carb dip for a day or two sometimes to clear out the passages when they get really bad.
The mix/pilot screw is about 2 1/4 turns stock, get it idling and let it warm up, then turn that screw in which is clockwise very slowly until the motor dies, then turn it back out counterclockwise one turn. That should be about perfect on them. The hanging idle once its warmed up could also be an air leak, i fought that for a while on one, it was the o-ring between the carb and the intake, since you had it apart and cleaned it, did you replace that o-ring? They get flat and hardened with age. There is also an o-ring between the intake and the head. The carb kits usually include both of those as they are generally a problem after removing a long time installed carb and they won't seal back up. Pretty common problem.
Also make sure the needle has the clip in the third groove and check the area of the slide where the stop screw hits it, a lot of people will adjust the idle stop screw with the slide rested on it, that will create a dent in the metal a lot of the time that causes it to hang a bit. A dremel tool with a cutoff wheel works well to polish the bottom of the slide notch smooth again and the just clean up the end of the stop screw if needed. What i generally do to adjust idle is give it a little throttle to bring the slide up off the screw then adjust it and let it drop back to idle, haven't dinged one up yet.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....