It helps, I think, especially if you really love those songs. Personally, it helped me a lot when I was learning English because I was passionate with the songs and thus easily understood and memorized the lyrics. If you love something, you get in your memory much quicker, and that's good. On the other hand songs and poems can only be a supplementary source for language learning because they have low content. The lyrics seldom get very long or complex, though some are poetic and beautiful. To learn a language seriously, you need a lot more than that. You should read pages after dense pages, and volumes and volumes, over a long period. There are so many words, phrases, and ideas in any given language that whatever you might learn from songs must be just a drop in the bucket. I think you should aim at bigger chunks for your main source.

It helps a lot in terms of improving your accent. You can also widen your range of vocabulary if the song have meaningful lyrics and words that can be used on a day to day basis :)

Songs in English are often helpful, assuming the lyrics are clear. It is particularly helpful in the area of word stress in sentences. The musical renditions of the Beatles and Frank Sinatra are among those recommended. Whether this method is particularly fast would be a matter depending on the learner.

Are Lyrics in English songs reliable for learning that? I mean are they always correct in grammar and meaning?

I suppose it depends on the type of music you listen to. I can't even understand some songs or the meanings behind the lyrics lol though I would say that yes listening to music is a good way to remember words and it can be fun too. Fun is important for learning a language in my opinion :p