You are correct about #4 being incorrect. The only time we'd use 'the art' is a set pattern of 'the art of X', where 'X' is whatever you wish to discuss.
Often, but not always, that construction is used to describe something we normally don't see as "traditional" art - 'the art of shaving', 'the art of war', 'the art of cheating', etc. You will of course see it used for things like "the art of Picasso', or 'the art of clay' - traditional art and artists as well. I seriously doubt you'd ever see it as the title of an academy, but I remain to be proven wrong.
As for #1-3, they're all correct and possible. I don't see much of any difference in meaning between the three. 'Arts' and 'the arts' by some definitions encompass a broader range of subjects, where 'art' is limited to the more traditional fine arts.
This Ngram seems to show them all about equally used.