Many composers today do not spend any significant amount of time looking at sheet music at all, actually.
This forum is a bit of an exception, but the general rule is most composers these days are using DAWs with a MIDI keyboard and piano roll view instead of notation view.
In fact, for the sake of composing, DAWs offer a number of advantages over traditional notation. You can see the entire score and what is happening in it in regards to vertical harmonic relationships without having to scroll or move your eyes much, you can make very precise timing adjustments, and you can play in the music in real time and edit the MIDI data afterward; it also gives you much more control over the "Mock up" performance of what the music would sound like if recorded live.
Depends on the individual composer. Most I would think have an idea in their mind, though.
I don't know any composer who does not play an instrument or sing. It would be very difficult to be taken seriously if one didn't.
Most are at least competent on piano, which has long been the composer's instrument of choice.
At the professional or academic level, composers often conduct their own scores and when doing so, they have the full score in front of them instead of isolated parts. So from the composer's perspective, it is mainly about keeping the ensemble in time, controlling dynamics and expression, and a reference to make sure the composer knows the orchestra or ensemble is in the music, and where it is headed so that he can cue in sections