"To write" or "written" is anything that is specifically written or typed using any kind of language. It can refer to music because music can be "written", especially song lyrics, which may be poetry. "Composed" is to construct or create, especially by combining or organizing things. It usually refers to some form of art, but in some sentences can refer to anything made using mixed ingredients. (Example: "The food was composed to many different ingredients.") In that sentence, the words "written" and "composed" are used to explain that the writer of the song's lyrics was one person, and the composer of the song's music was a different person. For example, you could say: "The song's lyrics were written by John Franklin-Smith, and the music composed by Abigail Adams." Or you could shorten it to: "The song was written by John Franklin-Smith, and composed by Abigail Adams."

To compose something is to create it from parts. In the context of music, the composer is the person who created the music. Usually composing music = writing music; but since the distinction is made it's exactly as Matt said. Some extras: 'Compose' often used to describe music, but not always. You can say 'compose' when you're talking about writing something (as long as you're not just writing randomly), but it's not often used. For example, you can compose an article or a sentence. It's not often used to refer to something that's not written. Here are some variations: Something's composition is what it's made from. For example, I can say that "concrete is a composition of calcium, silica, aluminium and iron", "concrete is composed from calcium, silica, aluminium and iron" or "the components of concrete are calcium, silica, aluminium and iron". All three have the exact same meaning. To 'decompose' refers to the rotting of dead animals. It does not take an object. To 'compose yourself' or to 'regain your composition' is to control yourself and make yourself calm, usually after some kind of emotional outburst.

Yes. The lyrics were written and the melody was composed.

The lyrics are the words (written) in as song and later (composed) with music which is melodious to listen.