It would be more helpful, if you would watch regularly movies/news in English or read books, newspapers, magazines. The text of many songs is very simple and have stupid meanings - with that your English would stay on a low level.

Yes it is an excellent idea, here are some songs and lyrics for you http://www.italki.com/knowledge/page/English%20songs%20about%20Love.htm

Yes and no. A well-written song will give you a good indication of how a phrase flows when spoken. On the downside: often pronunciation, grammar, collocation, decent vocabulary, logic, etc are ignored due to artistic license, or a lack of language skills by the writer. Also, since sub-cultures can sometimes be identified by the music listened to, I wouldn't assume a song written in English is a fair example of standard English. As a learner, it's often hard to work out which songs are good examples of the language. Definitely listen, but take it all with a grain of salt. ^^

If it's fun, why not?