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.

When I tried playing musical instruments as a teenager, I usually transcribed the music to my own notation. Something like "C D E F". Then I could play much faster.

In hindsight, maybe this was a wrong approach. Something like when people learn foreign languages the wrong way (most of them do), and instead of "foreignWord -> concept" associations they train "foreignWord -> nativeWord" associations. Then they speak really slowly because they have to translate twice "foreignWord -> nativeWord -> concept". The key to speak foreign languages fluently is to associate the foreign words directly with the concepts, skipping the translation to your own language. This usually also happens to people who learn languages the wrong way, but it can happen years later, and come as a surprise. Learning the right way, you can do it from day 1.

Analogically, if we tried learning music this way, we should skip the "C", "D", "E" sounds and try to create an immediate association between the shape on the screen and the position of the fingers. I have never tested this hypothesis, so I have no idea whether this is what skilled musicians do.

So I see two possible approaches here. They can also be combined: just because you may have learned to use some notation in a wrong way, it does not prove it is an optimal notation. The approaches are: 1) invent a better notation, and 2) invent exercises to associate the notation with movement on a level of reflex.

If I tried to invent a perfect piano notation today, it would probably be written top to bottom (instead of left to right). Because the piano keys are horizontal. A longer sound could correspond to a visually longer symbol; but the symbols would also slightly differ in shape (to make it easier to see how long exactly a long sound is). For example, there could be vertical lines corresponding to the black keys, small filled circles for short sounds, and something like filled letter "8" for twice as long sounds. That alone could be enough for a beginner.