That's pretty good. A couple points:
(1) Your description of breathing is not complete. The body has two mechanisms for breathing--the diaphragm and the intercostal muscles. Both work involuntarily, so the notion of "incorrect learned breathing" is quite dubious. I think it is more the case that people start to breath incorrectly when they think about it and start overriding their involuntary system. It's true that many people breath with their shoulders when they get exhausted, but often I think this is less "incorrect" breathing, than "breathing hard plus trying to maintain X," where X is a posture or something else that makes clamp down in their core. So, taking the deepest or biggest breaths is not a matter of choosing between the diaphragm and the shoulders, since the most expansion involves lowering the diaphragm and expanding the chest.
I was both a swimmer and a singer before martial arts, and I think I have a pretty good intuitive understanding of breathing and kiai compared with (a) other students and (b) my own understanding of other aspects of training. I.e., it is my comparative advantage, to put it in econ terms.
I think swimming, when done over a period of time and at a pace that keeps up the rate of breathing, is a very good tool for training breath control since the bouyancy of the water helps the body expand completely.
Swimming is difficult to do in the dojo, however. I would recommend people research exercises done by classical singers. Singing is not magic, and operatic singers do exercises to improve their control and the openness of their airways, which is another important issue that isn't addressed here, although I think it doesn't impact balance that much.
Another important point is that the diaphragm is not down around the belly button...
...and it moves basically down, not outward, so concentrating on pushing your belly out is not learning to breath with your diaphragm. Until you can coordinate all the muscles down to your anus, concentrating on feeling the area that we think of as "the stomach"--especially trying to feel inside the body, not the outer abdominal muscles--is more important than manipulating the bellybutton.
A note on kiai
--kiai is the opposite of deep breathing. It reverses all the muscles and goes fast instead of slow. When Payet-sensei demonstrates kiai in the kenshusei course, you can actually see his entire body contract and sometimes he moves off the ground. It's not just that you can see abdominal core squeeze under his gi (although you can see that, too) but even his arms and legs contract.
I have found for myself that on the days when I am lucky enough to be able to get into seiza properly, a good kiai in seiza feels like it almost lifts me off the ground. That is very difficult to do, however, unless you are relaxed. One way of relaxing is to get totally exhausted, and I have found that often my best vocal projection comes at the end of hajime geiko training.
This is not surprising to me since I used to do swim practice for a couple hours and then really belt it out in choral practice. Not always the right notes, but big volume!!
One training technique that seems to me to work well is to do bokken suburi and concentrate on doing kiai while making the cut with your back and the muscles between the bellybutton and the inguinal area
. This might be totally wrong, but it seems to automatically pull together your posture, projection of center, and exhalation.
(2) I understand walking a lot less, maybe even not at all. But I think your description of pushing off the back foot is wrong. I thought this was the key to suriashi, but when I started do it in kihon dosa, I was immediately told, "no, move with your hips." I haven't figured out yet how to walk using my hips first, but I think it has something to do with projecting your centre of gravity forward while coordinating the front foot at the same time so you don't get off balance.
(3) You've missed the thing that ties it all together, which is that learning breathing and walking are both fundamentally about learning to feel and manipulate your core/tanden/center-of-gravity, or whatever you like to call it. The other day we did an exercise in the kenshusei course where two partners stand facing each other and hands palm to palm at the sides of the body. You push on each other's hands and slowly go down into a squatting position while keeping the back straight, kind of like a deep front squat. At the bottom of the squat, you push on each other's chest to try to push the partner over backwards. At one point, I started to getting a little bit stable. It felt like somehow I was redirecting the energy going into my chest down into my perineum. I don't believe in ki, so there is a physiological/mechanical explanation for this, but I don't know what it is. Anyhow, the ability to use this part of the body is what I think breath and walking training are all about.