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.
My piano teacher usually hummed the first few bars of a piece to identify it when the title was not sufficient. When I try to emulate this with a new piece it’s much slower to go “note → D → D-sound”. If it’s a piece I’ve heard before, at some point my head fills in the rest without reading further, and then I only have to verify the score matches the sound in my head. This seems to agree with your hypothesis.
Top-down notation reminds me of vertical scrolling rhythm games like DDR and also Synthesia (though I have no experience with Synthesia specifically). My difficulties in learning to read piano sheet music were/are:
different clefs and clef changes
notes with many ledger lines
off-by-one-line errors (e.g. reading G as B)
following fingering recommendations
The last one might be entirely on me, I have a tendency to improvise fingerings and mess things up. But I think the other three are quite common problems to have. One of the things that trips me up with ledger lines is that the same note up an octave has different line/space parity.
My experience with rhythm games (and reading some sheet music rotated 90 degrees just now) suggests to me that fencepost errors are just as likely with top-down reading, and are greatly exacerbated by increasing the number of columns. I’m not sure it’s worth losing consistency with other music notation or western left-to-right reading systems. Maybe an anchor-note-position (say, a small mark on every bar line indicating C) would help with this, and possibly with clef changes too.
One other thing I used to not pay much attention to in piano were rests, and I would often hold notes until I had to play the next note regardless of whether there was a rest between. Playing rhythm games with hold notes has gone a long way in stopping me from doing this, as I have to pay attention to when to release the note also—extending the note-symbol to span its length could accomplish the same thing.