Re: kokyuho and the reverse breath.
I tend to see breathing exercises as a complication to movement. I think it can be as confusing as helpful and not many people do it well. Fewer explain it well. The idea is that your body contractions create respiration, like life support. Aligning your body movement with your body contractions maximizes power. There is a reason why your body "reverse breathes" after significant strain (such as running, working out or being active).
In the beginning, "breath exercises" are just simple tools to keep us from holding our breath or getting the wind knocked out of us. Belly breathing, deep breathing, whatever - these are breathing methods designed to improve oxygen intake and control a bunch of things triggered by stress like hyperventilation, fatigue, hormone dumps, etc.. Gracie BJJ has a lot of good info on this athletic breathing and it is how many athletes process oxygen. I think Rickson Gracie even has a yoga-style breathing technique. Ask a BJJ girl if she links her movement to her breath pattern. She will look at you like a three-headed cat because linking movement to breathing is bad fight science.
Later on, things get more complicated as "breathing" and "respiration" separate. Back to the Gracie-style, their breathing is pretty regular and not linked to movement. It feels more like a pressurized exchange of gases then our typical chest heaving "up/down" appearance. Sometimes it's easy to see how poor breathing impacts sport fighting. There are lots of good videos where you can see open-mouth breathing, mouth-piece breathing, heaving chest, rhythmic movement tied to breathing. My personal favorite is the ol' Hulk Hogan choke out arm twitch, but that is actually kinda real. All of these things show how poor breathing can affect athletic performance.