So why did Jack Frost have his power if no one believed he excisted?

Hmm interesting but he was also the only guardian that didn't lose his power after the childern stopped believing. I would say his power doesn't work like the others so he can't lose them.

"Some of the worst things imaginable have been done with the best intentions"

From numerous sources Jack Frost is some kind of spirit, possibly a shepherd of winter or THE winter spirit (I'm not entirely sure if his power comes from the Moon or Mother Nature), who just loves goofing around. Since his power source is nature based at this point, he doesn't need children's belief to fuel it, unlike the Big 4 Guardians who took the oath and then became their respected titles. That was why North asked him those questions at the end of the movie from the book and why his agreeing to those oaths made him an official Guardian.
In a nutshell: Pre-oath Jack is some type of spirit that enjoys having fun and Post-oath Jack is officially a Guardian who will (most likely) now be fueled by children's beliefs.

Pitch briefly mentions it in Tooth's castle. Only Guardians are affected by a lack of believers.

Then again, that poses the question of why Pitch lost his powers at the end?

He never officially became an guardian until the end of the movie. As others have said, the rules probably didn't apply to him like the others because of it.

I think that Jack Frost's power has been as a mischief maker who causes trouble by not being seen. This would need to change now that he has to become a guardian in the DreamWorks version of the myth.