To take the photo of the cyclist you should not have your camera set at Av as the camera will choose a shutter speed depending on the aperture you set. Since you are using a 580 flash, the camera will make sure you never go over the synch speed of the flash. Much better is to use Tv and choose a speed of 1/250th (or maybe 1/200th) and you get at least the max synch speed of the flash. This way the camera will automatically choose the appropriate aperture and the flash will give output accordingly. To be honest, I really don't think you will be able to get good results at a shutter speed of 1/200 for your pictures, and most likely will need something at around 1/500. I would go to ISO 3200 and drop the f stops and see if I can get a speed of 1/500. If I can't I might even go to ISO 6400 ... this all depends on the lens you are using, and the faster your lens the more leeway you get. Obviously by going higher than your flash synch speed that means that you cannot use your flash in normal settings, but the 5DIII can do quite well without flash.
Now to make it a little more complicated, if you really want to use flash and are willing to put the effort into understanding your equipment more the 580 has a High Speed Sync setting that allows you to overcome the sync speed limitation, this is usually used in daylight with high contrast subjects to equalize light, not sure how it would work in your type of scenario. Here is more info