How to choose a subject that is influenced by the music style you play ? Is there really a way
so that anyone can write lyrics ?
Also are songs* built on lyrics or lyrics built on songs

*By writting song I mean rhythm and melody of instruments
Anyone can write lyrics. As far as the other questions, there are really just no right or wrong answers.
There's no such thing; there never was. Where I am going you cannot follow me now.
Anybody can write lyrics, the quality will depend on practice and amount of reference of other lyrics

Now, the song -> lyrics / lyrics -> song debate is really easy: sometimes the song comes first and inspires the lyrics, other times the lyrics come first and the song is inspired by them. None of them is a good/bad option, however.

The subject is something you choose. You may choose a subject that is similar to those that your favourite bands use, but that's all to say about it.
Stand on your feet, partner. We've still got things to do.
All sorts of genres can include all sorts of lyrics. Lyrics are a form of expressing emotion, and is a piece that morphs into whatever you're performing. Rhythm, melody, types of instruments, all don't matter when it comes to songwriting. If you're singing a line you've made years ago and you regret about singing, then you're not writing your lyrics right. Don't fight the emotions just let it flow all around you...