What would the negation of this sentence be? "No me encanta tocar la guitarra."?
The "a mí" is optional, not required. It's only purpose is to emphasize the "me". "I love playing the guitar" (maybe implying "even if you don't). Nothing in the context of this sentence suggests that that kind of emphasis is necessary.
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Tocar is to touch and to play an instrument. Jugar is to play a sport, with friends, etc.
Encantar works a bit backwards, like gustar. The thing you love is the subject (here "tocar la guitarra"), and the person loving becomes the indirect object (here "a mí" and "me"). Since the subject determines the conjugation, you use the 3rd-person conjugation here.
To get the conjugation right, you can imagine the sentence with the more accurate verb, "to enchant": "Playing the guitar enchants me."
It shouldnt be. The verb can be translated ...correctly...both ways. Report it.