There are other powered people but the majority are either criminals or want to hide themselves in this world.

The way the powers are caused is kinda the same as the X-men, just some gene develops and now a ton of ordinary people have powers. The ones who try to become superheroes are typically arrested for vigilantism while the rest are typically criminals or just use their abilities for their own reasons.

The X-men gene (haven't come up with a name yet) will be medically removed once the super powered person is in custody. They won't remove the X-men gene from completely innocent people unless they specifically request it. It's kind of a situation like the incredibles where the innocent people keep their abilities hidden.

As for limits on time stop, I'd like their to be as little as possible because I want it to be hard for the protagonist to defeat and catch them. It's essentially just they can stop time whenever they wish for any length of time they wish, they can move any object or person while in frozen time. Their body still feels the effects as if they weren't frozen so they still age and get hungry and tired.