No
Magnets are not affected by rust coating them
If it was a problem they would have some sort of surface protection applied at the factory
Same story for coils. I have seen laminations fanned out from severs rust & the coils work fine
As for your mower, change your winter lay up routine
At the last mow for the season change the oil then wash the mower, remove the blower housing and leave it that way all winter
Firstly mice will not find the open engine an attractive winter address and if they can not afford to move into a better neighbourhood then you will see them so can evict them easily .
Finally fins should be cleaned annually and wires check for fraying & chaffing so you kill 2 birds with the one stone