On top of the opportunities it opens to you, it seems to make you develop some skills in your brain which make you become quite intelligent in other areas.

I think this too - I am guessing it is because each language is a pattern that affects the way we process concepts - having two or more ways to process is kind of like looking at something from several perspectives - but in this case, they are natively in all the positions simultaneously.

I guess the skills you develop when reading music and playing it makes your brain be quite open to many other intellectual tasks.

I think so. I wish I was skilled in music as it seems to leverage multiple brain areas.