If you write shippy fic, do you usually write characters who are either together or pretty likely to end up that way? Or doesn't that matter, because this is "just fic"?

To be honest two of the non-West Wing pairings I write the most for are both non-canon, and if the creators are to be believed, will never be canon. I write fic to "fix" that, so to speak. Fic fulfills two needs, in my opinion. First is the desire to see what comes next and/or what we didn't get a chance to see. The second is to play "what if" and/or explore things that canon cannot. Writing fic for non-canon pairings is no different than writing an AU.

If you write non-canon ships, what is it about the characters and what you've seen on screen that makes you think they should pair up?

It's different for every couple. There's just something about their interactions that's appealing. Truthfully there's no difference between shipping canon pairings and non-canon pairings. For a long, long time it wasn't clear that Josh & Donna would wind up together. We as shippers believed they would... but if West Wing had been cancelled earlier, they would never truly achieved canon status.