Ok, lets start with the smoke. The blue smoke is for sure oil. You might have a broken oil control ring. Time to invest in a new piston kit and have the cylinder checked and possibly bored out by a reputable machine shop. Shouldn't be more than ~$150. This would be my first priority. Search youtube for GY6 cylinder removal for instructions, they are very simple to work on.
As for the moving at idle and cranking, take a look at your CVT weights on the drive pulley(attached to engine) for debris or worn out weights. Sounds like it isn't disengaging properly. Likely going to need a set of sliders and a good cleaning.
The reverse issue may be related to the CVT, so start there and recheck it. The CVT needs to be in neutral for everything to mesh correctly and fall into gear.
Welcome, BTW, and take a look at our distributors here for deals on parts. Tom with SYC powersports can do your machine work if you don't know/trust locals.
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ASE Master and Toyota Master Diagnostic Certified
Buggy Building Trainee
'04 Dazon Raider 150 'modded'
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