Hi!
The AF functionality on the 5DM3 is sophisticated and takes a bit of experimentation. Here are a few thoughts.
Grab this guide linked below, and experiment. It is invaluable regarding all of the AF settings you have available to you. I've used it for sports (basketball, soccer), marching band, and ballet. Mostly it's been in very low light situations. This guide is excellent for choosing your AF points and cases.
http://cpn.canon-europe.com …rkIII_eng_January2013.pdf
I use only the cross type AF points. That drops me to 41 vs 61. Fine. Canon should have made them all cross type. They didn't. You can set your camera to use only the cross type. Do that. Also, you don't necessarily need a super high shutter speed. The photos below range from 1/200s to 1/500s.
Here are some shots I took. I believe for this I used AF point expansion with 4 points, Case 4, and AI Servo. I chose a group low in the middle, almost to aim toward the girls' feet.
AF setup is essentially the same as with the basketball shot.
Here, the football player was running directly toward me. I had to move. Here I used AF point expansion with 8 points, middle, center.
For the ballet photos, I used Zone AF point selection and Case 6. I believe in this case the camera will actually use all 61 AF points. For this photo, I used the right middle group, focussing on the girl in the yellow.
On back button focussing, don't bother. You've got plenty going on with the camera that the normal shutter button works just fine. If you've got things happening quickly, you want to react quickly. BBF sounds fine. For me, in practice it isn't fine.
One thing to note is that some lenses will not focus quickly. That said, I've used L zooms, non-L primes, and a Sigma ultra wide angle zoom and never experienced a problem with focus.
In the end, practice matters. The AF functionality of the 5DM3 is wonderful! Read, experiment, and repeat. Good luck!