I have this squirrel cage fan from an old microwave oven. It's the smoke exhaust fan and it's pretty nice. It is, I believe, an induction type motor but am not certain on that. I know I can wire it one way and get it to run at full speed. If I wire it differently it runs on low. Below is the schematic for what I've found. The left most schematic is the two ways I've wired it and gotten good results. The center picture is hopefully easier to read and the right most schematic is a wiring plan using a 3PDT switch, which I don't yet have.

I want your opinions on whether you think this is a good approach or not.

 

If it IS a squirrel cage motor, then it is induction style.!
Squirrel cage refers to the motor rotor.

 

Motor:
M25
91131
60Hz 120VAC
Some other numbers very difficult to make out
Amperage rating appears to be .75 (0.75A or 3/4A). Not big.

 

Reason for asking this question is because in the "LOW" speed mode power is flowing through the 7µF ±5% 400VAC cap. Will that harm the cap? The motor? Cause a fire? I ask because I don't want to do something stupid and set the house on fire.

 

If it IS a squirrel cage motor

It IS.

 

Ensure the cap is AC motor run-rated.
Usually oil filled paper, or similar quality.

 

Tis what it is

 

That appears up to the task!

 

Thanks Max. Preciate it.