I shot a lot of basketball games in the days of film when we didn't have high ISOs and good lighting so I had to use flash. The way that I did it was to set the camera on manual at the maximum shutter speed that the camera could use to sync with the flash (looks like 1/200 in your case) and at an f stop that would work to illuminate the subject at the power that my flash could produce at the ISO of the film. In your case, you might start at ISO 400, f5.6, and 1/200 sec. The flash should be on "auto TTL", if its an external flash. The duration of the electronic flash's flash will be something like 1/600 sec. or faster so the flash will top the motion rather than the shutter speed. Of course, because your settings are such that ambient light won't expose anything, you'll only have the player(s) illuminated in the photo.
I agree with the others that one shouldn't use flash at games because of the possibility of interference, etc. but, if you have to because your lighting and/or equipment can't support the available light option, then this is how you can do it with flash. Usually the players are so intent on the game that they don't notice the flash.

Suggest that you check to be sure that the focusing screen has not accidentally become dislodged and is preventing the mirror from operating properly. It happened on my 6D I. It is a user-replaceable part and you can re-install it yourself. To check, remove the lens and look into the camera body. You will see the mirror. The focusing screen is mounted firmly and flat against the bottom of the prism and should not be "hanging" downwards towards the mirror. If it is, you can push it back into place with your finger and it should lock back into place.

I've been using a Wein HSHSB hot shoe adapter with my Sunpak potato masher flash on my DSLR cameras. It appears that the current model number of the device is 990-560. It sells for around $50 new.

You can use Pentax lenses, both screw and K mount, on current Canon DSLR cameras with adapters that are available. They will only work in the manual mode (both for focus and exposure), which makes them inconvenient to use and hard to focus because the digital cameras don't have focusing aids like pentaprisms. Depending upon the Canon camera that you use, there may be an interference between the camera's mirror and the linkage that's on the lens that stops down the lens. I had that problem with my Canon 6D (full frame) but not my T2i (cropped sensor). I modified my Pentax fisheye lens to work with my 6D (cut the linkage off) and successfully used it for several years, but I had difficulties focusing it.

Have you considered using electronic flash? I used to take photos from the sidelines of college football games at night a long time ago using film and a camera that had an f 5.6 lens. I used a big electronic flash and had to watch for action that came fairly close to me to photograph. The electronic flash stopped the motion. There was no such thing as autofocus so I had to pre-focus for a distance, say 20 feet, and take the photos when the subject(s) came into that range.

I bought the Lytro Illum about a year ago. The camera is not a piece of junk, it seems to be very well made. It is more of a computer with a lens than a camera. Occasionally the firmware has a hiccup and the camera locks up. It is pretty complicated technology that is looking for a market. It has a sensor that has about 40 megapixels. As I understand it, each photo pixel is really made up of an array of sensor pixels, each with a different microprism in front of it. Each prism gives a different focus point. When you snap the shutter, you get several differently focused pixels for each photo pixel. The software allows you to post-process each image such that you can decide what you want to be in focus and what depth of field you want. The end result is a 4 megapixel resolution photo. The software does a lot of things that are unique to this camera (like letting you decide if you want your depth of field to simulate anything between f 2.0 and f 16) and the normal lighting adjustments. I use the Lytro software for its special functions then save the final image as a .tif file so that I can do the final edits with Photoshop.
The camera has tools in the firmware that help you adjust the focus and zoom to compose the photo that you want. All of your exposures will be at f2.0 but you can vary the ISO and shutter speeds. There are good training videos on a Lytro website. Taking a photo with the Illum is almost nothing like using a conventional camera. I bought it because the concept fascinated me and I had the time to play with something new. There is quite a learning curve to getting good results.
The camera system output can be a so-called living photo, a 3D photo, a still photo, and a video of an image where the focus point changes. There isn't much of a market for what comes out of the camera, as I see it. A small photo for a web page that has a lot of depth of field and was taken hand-held under low lighting conditions is one possible use. Like JWP I keep my Illum on the shelf and use Canon cameras to take photos, but I keep the Illum battery charged "just in case".