I suspect you should be aiming for shutter speeds of (ideally) at least 1/500, but with a slow max aperture on your lens that may not be easy to achieve. You may have to push the ISO pretty high, possibly beyond the limits of the camera. If you're maxed out on ISO and maxed out on aperture then you will not have any scope to improve shutter speeds. There may simply not be enough light for your gear.
What I would say is that noise is easier to deal with and preferable to blur, so keep raising the ISO until you can get shutter speeds that make the blur/shake acceptable. If the resulting files are noisy then hit them with some NR.
Find one of your photos that is well exposed without any editing and let us know what the focal length, shutter speed, aperture and ISO were for the shot. By all means post a photo (unedited and with EXIF intact) if you like, but from the exposure data given it should be possible to give more specific advice.
BTW, regarding the choice between Av and Tv, I agree it doesn't matter. You want the fastest speed possible (within sensible reason) so if selecting Tv mode you may as well go for 1/1000. In practice you will find that the aperture will remain wide open throughout the game. If you have "safety shift" enabled (I assume your camera has it) then the camera will lower the shutter speed as necessary in order to give you a correct exposure, but your aperture will remain wide open at all times. If you choose to shoot in Av mode then you will need to set the aperture wide open, in order to get the fastest shutter speed possible. At the end of the day the choice of mode between the two won't make a scrap of difference. The aperture will be wide open at all times and the shutter speed will be as fast as possible given the limits of the aperture and ISO selected.
Of course, if you are shooting in Av or Tv mode then you need to be mindful that the bright ice does not throw your exposures off, or that very different team colours (e.g. black vs white) have a similar impact. Personally I'd almost certainly shoot with manual exposure, locking my exposure to avoid these complications.