They actually teased the evil Mayor’s presence when Jimmy goes to the evil version of his school. There is a statue of the Mayor wearing Devil horns, which implies shocking scenes regarding him.
The planned 4th season was stated to be darker and more serious than the previous 3, so he wouldn’t be that humorous (or even humorous at all) if he were to make an appearance in Season 4.
I imagine that he would be a ruthless dictator in Retroville who may be on the same level of villainy as Evil Jimmy himself.
As for evil Betty, I would like to imagine that just like how regular Jimmy has a crush on Betty, the evil version of Betty has a crush on Evil Jimmy.
It’s possible that Evil Betty finds Evil Jimmy’s villainy to be really attractive to her.
I can’t imagine Evil Jimmy being nice to her though and reciprocating her feelings in any way, so in a way the relationship between Jimmy and Betty is again one-sided, just reversed on the Evil Earth.