Breathing and timekeeping... I want to say tempo and breath rate really aren't supposed to be connected. Is a drummer supposed to breathe in tempo to the song? What about odd timings? No thanks.
Hey, whatever works for the individual. I already have enough to think about without trying to match my breathing to the tempo. What's the point? Sounds like an over-complication to me. Tempo lives in the mind, heart and soul, not the lungs. Using my conscious mind to tamper with an unconscious process seems like a waste of resources. Just one guys opinion.
A relaxed state works best to me, which is taking in the air I need without thinking about it. Holding ones' breath runs counter to relaxation. So does humming along, IMO a big energy sink.
The only purposeful breathing that I do for drumming is for adrenaline suppression arising from stage fright, performance anxiety, or red light fright. I've not yet come across the need for endurance supported breathing, then again, I've never played under lights for 3 hours straight.
I did know a bass player who began his musical career on the tuba. He was fun to watch because he would instinctively breath his riffs.
I don't think we breathe in time, but breathing is important (other than for keeping us alive). When I first started playing out and was given a solo, I wouldn't breathe, and I would fill the space up with way too much stuff. Finally my teacher told me to relax and breathe, and that would allow me to take the time to think about my playing.
As long as you are relaxed and not holding your breath (or hyperventilating) you should be ok.