Probably not. You definitely gain high ISO image quality and image quality in general with full-frame (a much larger sensor), but the most recent crop-sensor cameras have significantly closed the gap. Image quality is more a function of lenses and skill anyway. And the differences are much less evident with video if that's important to you.
A potential negative to full-frame is that you lose the 1.6x crop factor (which gives the perception of adding 1.6x to your focal length with a crop sensor camera). Technically the image is the same, but that's the gist without overdoing it.
So your 200mm lens on a crop sensor camera will seem like a 320mm lens whereas it will seem like a 200mm lens on a full-frame camera.
You also gain some depth-of-field capability with full-frame, but I find that to be a total non-issue. You don't necessarily want a razor-thin depth-of-field (unless you're being ultra-creative and precise) and definitely not for "running" photography.
If you're to split hairs though, as I often do, you may wish to consider a full-frame camera for the opportunity to achieve the best image quality possible in all conditions. You'll pay more for it and the lenses (in many cases) and experience an added limitation or two, but it can be worth it if you're a hair-splitter.
Do keep in mind, though, that for outdoor sports photography it might make almost no difference in image quality because your ISO is almost always very low or reasonably low. Your pictures tend to get grainy with a crop-sensor camera as the ISO increases, but usually not a factor outdoors and less of a factor now anyway with the newest crop-sensors.
I purposely left out an 85mm lens because that duty is often handled by the 135mm lens and stepping back (or even the 100-400). If you must though, all things considered, I do gravitate towards the inexpensive and outstanding Canon 85 1.8.
As always, anytime -
This is the entry-level version from Canon. Features a much improved focusing system over the original and has also added a swiveling screen.
The image quality is no different than what the more expensive models offer.
I'm a big fan of simple and high quality and this would be it.
I normally favor prime lenses for their arguably better image quality, but the versatility of this ultra-wide-angle is a very acceptable compromise (with more than adequate image quality nonetheless).
f/4 is probably as low as you'll ever need to go in this range and image stabilization can be nice unless on a tripod.
This lens may be worth considering if you wildly appreciate the highest levels of that elusive "professional" color and contrast.
Some criticize this lens for lacking sharpness, but photography isn't all about sharpness. Canon could make it sharper, but then it might lose that "look".
A good way to achieve "that special look"; with the distance allowed by this focal length creating a very natural/non-distorted result. Universally recognized as among the best.
IS isn't really needed in most cases as long as you keep your shutter speed around 1/160 or faster (with 135mm).
It's a tough call between this and the 100-400, but this is a legendary lens capable of extreme excellence and 300mm is typically versatile enough for a telephoto lens.
For sports I would normally turn off the IS, keep the shutter speed at 1/320 or faster and use good technique.
The 400mm 5.6 prime lens also works.
This lens is overwhelming versatile and offers remarkable image quality.
f/4.5 at 100mm isn't great, but since the DOF decreases with an increase in focal length f/5.6 is pretty good at 400mm.
I'm going to give the nod to the 300mm prime lens (at half the price), but this lens may serve many more purposes.