Computer software is just a more modern way of exporting the composition than pen and paper. Often more convenient too, with selective cut and paste, automated transposition, short cuts to enter repeated notes ,etc. There can be problems too which require some experience of the compositional software, like bars with dual metres, ornamentation which leaves 43 demisemiquavers in a bar, etc. I wouldn't compose with a computer per se but only ever use it in the secondary or even tertiary stage of getting the stuff formally notated. In short, the computer (to me) is only a secondary tool to facilitate getting the music in a publishable form and not a primary method of composition.
Ted is right in drawing attention to the issues in notating improvisations; I doubt that it is always feasible, or even desirable, to attempt absolutely precise rhythmic notation of filigree, for example.
To answer, personally speaking, one question raised, I have no idea at all when it first was that I could read piano music in my head and play it back, but I can't remember a time when I couldn't. (There has to be a small caveat in that I'm far less capable at playing back multi-instrumental polyphony in my head.)
Interesting to hear other members' compositions. The Polonaise reminded me of some Frank Bridge and similar music.
This is one of mine, and began life as an improvisation - I removed the bits I didn't like, improved the bits I did, and shifted around the various remaining segments until I liked the dramatic effect, writing linking passages where needed. I can assure you it took forever to notate with software

(and even then the notation isn't absolutely 100% accurate in terms of correlation with what I've played..)