Based on your poll options, I'd go w/either 1d Mk III or the 7d. When shooting sports, I want 4 things; a high frame rate (8+ fps), good/great AF performance, ability to shoot at high ISO (ideally 6400+, at least 3200), and fast settings adjustment (especially focus points).
I own a 5d II and although the high ISO performance is pretty good, the AF performance and the frame rate is lacking. For that reason, I never use the 5d II when I go to the gym (or even the ball field).
The 6d, although a great camera, does not have the "joystick" multi-controller. I've become so accustomed to having this control that the lack of it has become a complete deal breaker for me. I love being able to use the multi-controller to change focus points on the fly and would be totally lost without it. For this reason alone the x0d series (ie; 70d) and 6d have never made the cut when I've been buying camera bodies.
The 7d is a great camera body and, actually, is usually my go-to sports body. I've owned a 7d since they were first released and the only complaint that I've ever had with it is the high ISO performance. Out of the camera, when shooting at ISO 6400, the camera does exhibit noticeable noise. Nik Dfine and the Photoshop CS6 RAW converter (noise reduction component) have become my greatest alley in these situations. If I can, I try to keep ISO at 3200 when shooting w/the 7d but some gyms just don't allow this (FWIW, I step my ISO in full stops... I don't like my camera body pushing exposure one way or another to produce the fractional ISO stops. I can do this in Aperture and/or PS). The 8fps of the 7d is awesome (I use a Lexar Pro 1000x UDMA7 card and it keeps up nicely) and the AI SERVO focus system works great as well. Overall this body has served me well.
I think that the 1d Mk III would be an excellent choice as well. If I only had one camera body and it was between the 1d Mk III and the 7d, I might lean toward the Mk III. The Mk III has a 1.3x sensor as opposed to the 1.6x that you'll find in the 7d. This would be an advantage for me as the 1.6x is a little "long" for my 70-200 2.8 IS II lens in small gyms. I would also expect the Mk III to exhibit less noise since your individual pixels on the sensor are quite a bit larger (10.1mp on a 1.3x vs 18mp on a 1.6x). You can also push the ISO to 6400 with expansion so you'd be at least on par w/the 7d here (although the 7d will go to 12800 w/expansion it gets really noisy so I seldom use it). The Mk III will also fire 10fps and has dual memory slots (a big perk if you ask me). On the other hand the Mk III sports dual Digic III processors vs the dual Digic IV on the 7d which might give a slight speed advantage to the 7d. Without having the two bodies side by side though, I don't know if this would be perceivable or not.
I think that the biggest reason that I own a 7d instead of a 1d or 1ds is that I also own a 5d II and 5d III (and XSi). Being able to share batteries has been a big advantage for me and the 3 bodies actually compliment one another very well. The 5d III has proven to be a great sports performer as well. The FF sensor and super high ISO performance of the 5d III has more than justified the slower frame rate (compared to the 7d) in quite a few situations (the aforementioned small dark gyms especially). However the 5d III is significantly more expensive so, unfortunately, not an option for you at this time (although it will be well worth it to keep an eye on the used market, in a year or twos time, who knows?). I've strongly considered swapping on my 7d/5d III combo for a 1d X but doing so would bring another battery type into the equation which is something that I want to avoid (when we go hiking my wife will carry the 5d II so the LP-E6 is here to stay even if I upgrade my bodies).
Anyway, in your situation I think that you'd be more than satisfied with either the 7d or the 1d III. Based on your lenses however, the 1.3x sensor in the 1d III may prove to be a great advantage in small gyms (although conversely a disadvantage on a large field, which you didn't mention that you do). The 1.3x will also be an advantage for family shots when you tend to want to shoot wider. On the used market, the 1d series bodies are NEVER a bad buy either. In the end, I guess that I'd lean towards the 1d III. Find one that has been taken care of and has a decent shutter count (<150k) and you should be good to go. I bet that if you scour
POTN for a few weeks, you'll have no issue finding one that will be exactly what you're looking for.
