Electronic – Why is there a resistor inline with switch in blender motor

You say that the motor draws 350 mA. If you pass 350 mA through your 220 ohm resistor, Ohm's Law says there will be a 77 volt drop across the resistor - since you only have a 3 volt battery, this obviously won't work!

Ohm's Law indicates that 3 volts across 220 ohms will only result in a current of 13.6 mA.

Common LEDs typically have a maximum current of 20 - 30 mA, so they cannot be connected in series with your 350 mA motor - the LED would vanish in a puff of smoke!

If the motor is designed to operate on 3 volts, it should be connected directly to the 3 volt battery, with no series resistor.

No, there isn't a simple or safe way to reduce the frequency of the supply.

I doubt that a small high speed blender would use an induction motor, it is much more likely to be a brushed series wound 'universal' motor. See if the armature is wound, if so, that's what it is.

These are usually controlled for speed using a phase angle controller, basically a heavier duty version of a lighting dimmer switch.

You have several options.

A Van de Graff uses very little power, almost all the energy is spent in roller and belt friction, a good mechanical construction should keep these losses reasonably low. Try to get a small DC motor, and run it from batteries.

If you must persist with mains, then put a mains filament lamp in series with the motor and see what you get, if you can't get hold of a phase angle controller rated for the power of the motor. This is perhaps the least dangerous way (note I don't say safest, it's only as safe as your competence with making sure mains connections are insulated) of messing with mains. Use more lamps in series or parallel, or different values to adjust the speed.