A - I also use scale degrees for this. I'm not fast enough to sight-read and transpose at the same, but I sometimes switch to intervals when transposing a single melody line. Most of my experience transposing and playing comes from trying to play songs on Bb trumpet written for concert pitch. I had some note-to-note mappings memorized from frequent use, but not anymore.

B - Definitely plausible to me. Something along the lines of "when all you have is a hammer" and certain modes of thinking being more useful than others for certain skills. I don't have any particular examples, but I think in math there are quite a few abstractions / extensions of lower level concepts that are a lot more 'natural' if you think about them in a certain way.

C - Can't think of any.

D - Agree with Anomylous on this one.