Your mixture screw may be covered up like many are . Luckily mine was not and the idle screw is so big that I can adjust it with my fingers . When seasons change you may have to adjust the idle . Always let it warm up by riding around a bit before adjusting the idle . When it is cold sometimes mine will idle high for a minute or two then idle down on its own .
Should not turn the wheel on the CS at idle unless the choke has it running at a few more RPMs . Sometimes mine would want to want to spin the wheel but not a full engagement . Now you can get stiffer clutch springs like the 1K , 1.5k or 2k , which these engage at their rating that much above the stock springs . 1k the clutch engages at 1,000 RPMs more than the stock ones . 2k may be too much as it will wear out the parts faster and can have you popping a wheelie . I use 1.5k since I wanted to try 8g sliders to reduce the RPMs at cruising and top end . Most use 4g . I had 5g when it was a 50 but with a BBK it was too light . The 8g only slowed down the takeoff as I can still hit 9k RPMs going up a steep hill with a running start . So it looks like I will have to get taller gears as I want to tune it for long rides and hope to get better MPGs .
See what size idle jet and venturi you have and try a size up IJ . Mine had a 40IJ and I would have to give it throttle to start and during a long idle as I had to adjust the idle screw every 2 months . With a BBK it got worse even though it ran great while riding . I went to a 45IJ now it starts with no throttle and idles perfect . It was easy to find main jets as many use Mikuni VM11/22 but my TK carb had an oddball IJ and I only found it at one place . It does cost a bit retail but worth it . If you can get or know someone with a commercial account it will be much cheaper . Here is the IJ if yours has a TK (Teikei )
http://shop.jupiterscooters.net/Idle...MP-JET45TK.htm
I also am enjoying the 47mm BBK from MODCycles with 10mm wristpins . Looks like the Hoca kit , has way larger ports than the 50cc did along with a full ported exhaust port . The 50cc was shaped like a D instead of an O , so it restricted it from getting high RPMs . 50cc would max at 8k and if I went higher than that downhill it would fizzle and stall . Even a racing CDI made no diff on top end RPMs . With the BBK , Uni and a racing pipe with no baffle it can not hit 10,500+ but I try to not go above 9k to be safe .
2Ts will always need to warm up prior to riding , more so in winter . I rev it up just a bit on CS which also warms up the belt too . Only rev it up just enough to get the tire spinning and NEVER race up a cold engine . A NGK iridium does help it start and run better than a standard plug , but will not make it go faster . I use NGK BRP8HIX since it is a cooler temp plug than the BRP7HIX which some in cold climates may use in the winter . Winter seldom gets below freezing where I live