how do they work and how do i get one? could another player do it for one of my pets, like the infertility thing? or is it only like a one shot thing

It depends on which one you use. There are two that I know of- the Lesser Genetic Mutator and the Greater Genetic Mutator (There are also trait disruptors, but thats a different thing altogether.). The lesser one is found in the Scale Shop, and gives a pet a random carried or visible trait. The greater one, in contrast, can only be bought from the Diamond Shop (or traded, bought via auction, etc.), and allows you to choose a trait, whether its carried or visible, and what slot you want it in. Both of them can only be used on tamed pets, and the pet in question can only be altered by them once.

there are also Lovely mutator and Lucky mutator, only buyable from the diamond shop during the Festival of Love and the Fortune Festival respectively.

Lovely mutator:
Can give any pet the Restricted Trait "Valentines Lace".
Alternately can give a pet Lovely Hearts mutation if the species has it.

Lucky:
Can give any pet the Restricted Trait "Lucky Dust".
Alternately can give a pet Lucky Runes gene if the species has it.

outside of the commons mutators there are also restricted traits mutator that work on any pet but replace one non restricted visble trait into a rendom trait from the selected trait set. You can get those from the diamond shop during festivals and events.

Another way you can get mutators is by going in an activity like the Winter worderland (for winter) and the haunted house (automn). It's a realy rare droop from the third level.

those are Disruptors, not mutators, and function completely differently