>Does listening to music in the language help with learning the grammatical parts of the language?
Yes, but you have to take care in choosing which music and which band you are going to use. If you don't truly like the music you won't want to listen to it over and over again, and if you choose a band with lyrics that are too basic or too nonsensical you either won't learn much or you'll waste your time with something less than useful.
Sometimes you'll also find good songs that repeat certain grammatical patterns, like this one by Jaurim (Korean band) from their first album:http://www.pagef30.com/2009/11/learning-korean-...
This song has a lot of ~도 (even if, like Japanese ~ mo) as well as 지 않아 for forming the negative. They are:
손을 내밀어도 말을 걸어봐도
왜 나를 사랑하지 않아
곁에 다가가도 미소 지어봐도
나를 바라보지 않아 나를 바라보지 않아
너를 바라봐도 좋은 선물해도
나를 알아 주지 않아 나를 알아 주지 않아
A lot of other grammatical patterns too, like quoted speech (the 고 in 그녀가 말했어 내가 널 잃어가고 있다고) and relative clauses (아름다운 너에게선 - from beautiful you, or from you who is beautiful), plus some other things.
The best part about using songs in learning languages is that after you are done listening to them they stay in your head for hours and hours.