Great questions! All the big stuff. I really wanted to respond to the musicality problem, because for me this was the hardest aspect of playing. I always had the fingers and the reading, but for most of the time didn't have teachers, so no one ever "called" me on this. I was very quiet and static with lots of pedal. It was a conductor I started playing for in recent years who's really into baroque music with a lot of (unnotated) nuance, who made me do all this extra articulation and got me thinking about making different shapes and sounds on my instrument. This is who helped me the most-- in other styles as well (I was so inspired about music again that I finally started taking lessons with a fantastic pianist-- it's never too late). I believe that many people who sing and play unmusically, although they sound cold, may not be cold people at all, may in fact have a great deal of feeling for the music but haven't learned how to express those feelings within a musical vocabulary. Like speaking a language you can get around in a little, wanting to say something really personal or specific and knowing a similar word but not the exact word you need. Some people know instinctively how to show feeling in music, and others, like me, need to be taught the specific musical gestures-- shapes, articulations, pedaling, better voicing, clearer phrasing, whatever. My conducting friend isn't a pianist, but was able to explain to me how he makes musical decisions, and he talked me through all my solo pieces for a long time until I got to where I could put it together for myself. It felt artificial at first, but it trained me in the habit of looking for artistic things to do, and now I do a lot of detail even when sightreading, because it's so ingrained. So I wanted to encourage you in all you're doing to help your own students with their musicality. I would bet that the specifics you have to ask for now will become habits as they go along. And if they end up with your interpretation for now, they have a good model for how music "works"; I think that's fine. It sounds like you're doing great.
The issue of transfer students with bad habits is tricky. I use the repertoire and the next level of playing as the excuse for changes. If they're breaking a joint, I explain that they can't do that in these new pieces because they won't be able to move quickly enough. That always hooks them, because everyone wants to play faster, and I didn't have to malign the other teacher. Or at the next level of playing, we can't write in the note names anymore because there are too many notes now (I look for thick chords or long busy runs to lessen the temptation). Just my own path through those particular landmines. For teenagers or particularly bright younger ones, I explain the physiological or artistic reason behind my advice, making no reference to the other teacher, only talking about the goal at hand.
You sound like a great teacher. Where were you when I was growing up? Probably not born yet... anyway, best of luck.
Topic: difficulties in teaching