My process is not fixed in stone. There are times I wake up, still with a dream of a particular sound and vibe. When that happens, I sit down and create it. I've written a couple things like that recently (it may start with a feeling and then it flows... One progression to the next). I often hum an idea and record it into my iPhone so I don't forget it if I'm driving and don't have access to an instrument.
At my home studio, I sometimes write by jamming and experimenting with different tones, patches (on my keys) etc, and when things begin to flow and eventually tie together, I record them. I write on keys, bass and guitar, but primarily on keys.
The original band I play in is a three piece. We recently just reunited after a 24 year layoff. It's so nice to be cromposing again and it's absolutely the most inspiring thing I'm doing these days. All three of us compose on many different instruments. The style? A mix of rock, prog, pop, classical, psychedelic, etc. As a group, we often compose together. Sometimes one of us or all of us will bring in a just one simple part. With the help of the other two, that turns into a fairly well composed song after experimenting and jamming through different varieties of progressions.
I love that process of creating a composition with my dear friends.
Other times, one of us will bring in an entire song and the other two add their individual instrumental parts to it sometimes as directed but always with an air of openness to the other's musical strengths.
The guitarist and I both use Logic X to compose on and we also use it live with Mainstage to control keys and prerecorded backing tracks.
As far as lyrics go, it's very seldom that a lyric gets written first. Tone and phrasing of chords has a lot to do with the choice of vowels sometimes. That's primarily why I finish lyrics last.
As far as lyric style... I hope at some point that at least a few of my lyrics will be as powerful as Peter Gabriel's.