Sniper, as I mentioned on page 1, I've had similar issues dealing with low available light. The difference was that I was making different choices than you, but coming up with similarly dissatisfying results.
The lighting is pretty similar to yours. You mentioned 58mm, f/4, 1/50, ISO 10000. Mine was 35mm, f/2, 1/200 or so, ISO 10000. ..about a one stop difference.
I decided to try the 550EX flash I had lying in box, unused for ages since I bought a fast prime (35L). Took a couple of shots last night and went, holy...umm...cow (except it wasn't cow).
Same basic lighting as before, but this time:
5D3, 24-105L @ 75mm, f/4, 1/200, ISO 400 (Manual mode with auto ISO)
550EX on hot shoe, bounced up and right
IIRC, nothing done to RAW except a tighter crop, and maybe a small white balance adjustment (I don't recall). The RAW is pretty darn sharp but might not have downsized well since I didn't apply any sharpening.
The point is to show you a result in similar light with a flash. The difference is that there was not much motion in this shot, but still the point is that the shutter speed was 1/200 at ISO 400 as opposed to your 1/50 at ISO 10000 without flash. You could choose to go faster than 1/200 or higher than 400!
edit: see picture in comment #52 below.