Hi Kevin,
Great shots!
I've just done what you're considering so I'll give you my initial thoughts:
Obviously everyone has different tolerances for noise levels, but the 5DIII produces much cleaner high ISO images, no doubt about it. I think ISO 6400 will be my new "normal" while shooting in low light as the NR required is quite minimal IMO. If I'm shooting strictly for the web, I'll have no problem shooting ISO 12K. Big advantage 5DIII....
The 5DIII has the better af system. Dealing with the 61 focus points can almost seem overwhelming at first but the flexibility of the system is incredible. The 5DIII locks faster and is more consistent, it's essentially the same af as the 1DX. I expect my keeper rate to go up, not way up since the 7D af is pretty darn good in it's own right and I was quite comfortable with that system. It was groundbreaking for a Canon APS-C body back in 2009, but technology has moved forward. Advantage 5DIII....
Going FF, you're also going to need a 16-35 or 17-40 to get as wide as your excellent 10-22, something to think about if you shoot really wide. Your long glass is also going to seem shorter, I won't say APS-C gives you more "reach" because that's a holy war around here..lol
The 5DIII shoots 6fps vs the 7D 8fps, I haven't tested this in any sports yet, but the difference is quite noticeable, this may or may not be a problem as I'm quite used to the performance of the 7D, I'll definitely need to improve my timing to compensate, the 7D also has a larger RAW buffer. Advantage 7D...
Adding the 70-200 2.8 will give you another stop for low light which might be all that you need to keep your ISO at acceptable levels. Having said that, I have no problem shooting at ISO 3200 with my 7D, especially for web output. Yes, it takes some work in PP, and everyone's tolerance for noise is different, I have more than a few prints at ISO 1600 that came out quite nice with the 7D, mostly 8x10 or smaller but if you're printing, noise will be less of an issue than you think. However with the 5DIII, I have no doubt I'll be printing much larger now and probably up to ISO 6400. 
The 5DIII raw files are IMO more "durable" they require less work to get good results and can handle a bit more aggressive PP when you need it. Your PP skills might be better so it may or may not be an advantage for the 5DIII.
You might also want to check out POTN member Team Speed's threads on the 7D, he's able to work magic on the 7D from a noise perspective:
Mini-Review: Trying To Get the Most out of the 7D High ISO Images
Sorry for the novel 