My fFavorite Musician

In a title all primary words are capitalized.

I like OST musics of movies, games and animations.

Music is a field of interest, and therefore we don't pluralize the word. However, movies, games and animations are particularm discreet items so we use plural forms with all three words.

One of the most impressive musicians is Ennio Morricone.

"One of the" indicates that there are many musicians, so we pluralize the word musicians to show that. One of them is Ennio. "One" and "is" agree in number.

I remember hearing his name all the time in my youth, when I enjoyed movieswas a movie buff. OR... when I was actively interested in film.

The statement is correct as you have written it but it seems a bit awkward to my ear.

ByIn the movie, "Mission," movie, I heard a heart-stopping musicsong (or piece) called "Gabriel's oOboe"."

The use of commas is guided more by convention than by rule. The convention is to separate a title introduced in the middle of a sentence by commas. The word "music" is not used with the indefinite article "a" because it does not refer to the particular instance, but rather a universal concept. You would use words like piece, song, or work. to describe a specific instance of music, using words like "piece," "work," or "song." You could have also written:
In the movie, "Mission," I heard "Gabriel's Oboe." The piece was heart stopping.

I think Ennio Morricone's music has raised movies to a whole new level.

I would use the present perfect tense rather than the simple past for the verb "to raise" because The present perfect tense is used to talk about events that are relevant to the present but that happened in the past. It is usd to talk about an action that started in the past, without mentioning a specific time. Both are correct but I don't believe you want to to note the influence of Morricone as a historical past, something that happened and it is over now, but rather in my mind you want to tell everyone that the change goes on into the future.

My favorite Mmusician

I like OST musics of movies, games, and animations.

One of the most impressive musicians is Ennio Morricone.

When I watched "Cinema Paradiso", I was impressed by the music rather than by the movie itself.

ByDuring the "Mission" movie, I heard a heart-stopping music called "Gabriel's oOboe".

It is the music that makes me remember those iconic scenes as if I saw them yesterday.