I believe the heroes are supposed to be teenage versions of those heroes (this would be why Wonder Woman seems to have actually been in a relationship with Rod, why Batman doesn’t have an actual batmobile, etc.). I think the idea was to make an intentionally silly/campy setting, so we aren’t supposed to be worrying too much about things like why Clark and Diana would be going to a high school in Gotham. So, rather than being heroes going undercover as high school students to uncover a crime ring (as many viewers probably suspected at first), I think they’re just supposed to be four teenagers with superhero/villain alter egos who coincidentally were all going to high school in Gotham and got caught up in this plot. (I believe the GM said Catwoman knew the principal was up to be something and was trying to uncover it, but Clark seems to have just been trying to be a good student.)
On DC Daily they were tossing around the idea that the whole thing might be some kind of virtual reality/psychic projection that the characters are trapped in, which is still possible. But I think it’s just supposed to be an intentionally silly premise that isn’t meant to adhere to any comics continuity, and that doesn’t really bother me; it’s still fun.
I also agree that Xavier isn’t really suited for Batman. I’m glad the GM at least cut off his hero points for trying to kill. (Though of course it should be noted that the very early version of Batman killed criminals without a second thought.)