You'd need a ton of 18650s to power a minibike. Lead-Acid lawnmower batteries would be more cost effective.
Drill press motors and alternators are driven by AC (actually alternators generate AC, but they're pretty much the same as a small 3PH synchronous motor). A controller could drive an AC motor but the unpredictable load of a minibike would require the controller to be closed-loop, meaning you'd need to add a hall-effect sensor cluster to the motor. That can get expensive and complicated and in the end, you just made a brushless DC motor anyway. Stick to electric scooter motors.
I suggest you look into Andymark. They have a sweet DC motor controller:
Victor SPX Motor Controller (am-3748)
Gearbox:
EVO Slim 3 CIM
And motors (you need three motors, each with their own controllers):
2.5" CIM Motor (am-0255)
If you want a possibly more powerful system, check out their Redline motors and Redline 4-motor EVO Shifter.
In the end, considering the expense and the power output, I still prefer gasoline engines. But hey, I'm happy to play with other people's money!