I don't think she is - a lot of places in town disallow pets, and I'm sure the school is no exception. Mona's just really good at sneaking in Fang. Also, it's possible people might be a little lax because they stopped caring about it after a while.
Some of the villains' defeats elude me. We see how they'd lose in the imagination spots, but how would they lose IRL? For example, Mona defeated the Sun Goddess by reflecting her rays back at her and destroying the tanning machines. How would this work in the actual events of the series, when the villain didn't have rays like in the imagine spot? Moreso, a tanning booth is on fire.
If Mona's aunt is a clue (I don't really remember why, they'd get too freaked out by Mona and just simply distance themselves from her.