Hola Malak. La clave es que no todo es sintaxis. La oración "Me interesa la música" es perfectamente correcta. Pero puedo suponer que la persona que está diciendo esto intenta reforzar la idea. Por ese motivo dice "A mí me interesa la música", la cual es una oración con más fuerza. Las dos oraciones son correctas, pero una es más potente que la otra. Con respecto a "estoy interesada en la música", es una oración en voz pasiva (I´m interested in music). Al contrario de lo que sucede en inglés, en español solemos usar más frecuentemente la voz activa (Me interesa la música). Espero que te resulte útil. Saludos desde Argentina.

"me" is not optional in that case, because interesar is a reflexive verb and it requires a reflexive pronoun: me interesa, te interesa, le interesa, etc.. "a mi" is optional as it has been already mentioned. It is really very common to use "a mi" in that case, even if it is redundant. A real situation in which "a mi" has a real meaning would be the following: a group of people is talking about what they are interested in. Let's say that one of them is interested in music and the others are saying that they are not interested in it. In that case this person would say "a mi (sí) me interesa la música".. It's like "I DO have interest in music".

I believe that the phrase is in the context of a conversation, they are talking about their preferences and that person says: "A mí, me interesa la música" I hope I have helped you