AC induction motors will work just fine with variacs. I know this for a fact because I once worked for a manufacturer of AC induction motors. We used variacs all the time for powering and testing the motors.If it is not a universal motor capable of running on DC it will not work.
No way. A huge advantage of AC induction motors is that they regulate speed quite well at a set line frequency. Any induction motor I have seen that is run at line frequency (60 hertz) and reduced voltage will draw more current and heat up. The motors I have experience with a multiple horsepower, not a fractions of fractions like a small fan motor.Edit:
AC induction motors do not speed regulate very well without a variable frequency drive. Depending on the motor, it may be possible to get some speed regulation, but it will be inefficient and the motor may get hot.