I think it generally depends on who writes him. Mxy and the fifth dimension folks tend to have undefined powers by their very nature. In theory, they could be all-powerful gods, but in practice, a writer can do whatever they want to with them. This applies to all superheroes and villians to an extent, but with the wonky nature of Mxy and his ilk, I think it's more so.
The vast majority of the time, Mxy is only ever used as a joke character. He is rarely ever a serious threat so we never get an accurate view of his true power. One time he was used seriously was during the Emperor Joker storyarc. Joker had managed to steal a fraction of Mxy's power, and then Joker was able to recreate the entire world in his own image. Joker tortured and killed Batman every day just to resurrect him the next day and do it all over again. The Spectre tried to stop Joker and failed.
in Alan Moore's Whatever happened to the Man of Tomorrow, Superman traps him between the Phantom Zone and the 5th Dimension, effectively killing him.
In the New 52, Action Comics #15, Mxy pulls the entire DC Universe out of his hat for the 5th Dimension's King's amusement.
Morrison is being quite faithful to the original stories; Mxy from the start had omnipotent powers and could do whatever he wanted in Superman's dimension; the thing is that he's a jester in the 5th dimension and he doesn't really have a mean streak, although he considers for two seconds to make himself ruler of Earth; but pestering Superman is a funnier career move. Those Mxy stories from the '40s are just hilarious, almost as insane as Jack Cole's Plastic Man.