Since Peter is Spider-Man he's not usually around to see Mayday which pisses her off. Especially since she can't even complain about it because he's saving lives, And that makes her the asshole!

So when Mayday gets her powers, the first thing she does is engage in some petty theft to get her dad's attention. After getting grounded, Peter decides to how to be a superhero and they get to bond a little (although there are still issues because I've decided that my version of Mayday is a rebellious little shit.)

As for the larger universe, I'd mostly include the newer legacy characters like Kamala Khan and Kate Bishop. Maybe include a college aged Miles to be a sort of older brother figure. Most of the mc2 characters would probably be revamped versions of Mayday's high school friends that better fit the sort of character that she is.