Thanks for all the feedback!
Regarding setting the alpha on the whitespace: I tried that in the Unity Editor and faded the guy out of existence. So that was a no-go. I might need to learn something other than MS Paint to pull that off.
I am currently trying to work on the "moving at supersonic speed and getting stuck in a wall" bug, but am...uh, hitting a wall. I'd encountered this bug in the Unity Editor, but after simply restarting the Editor it stopped bugging out. So I shipped it as it was, and hoped nobody would be speeding through walls. I actually cannot reproduce the bug when I play the game on my own page (the guy goes as intended), but it does bug out in the Editor after test-running more than once. So, while I may be unable to fix the some-people-go-too-fast bug, I should at least be able to keep affected players within the game boundaries.
Patch is ~~on its way...~~ up! **A slight fix has been implemented which will hopefully prevent moving-too-fast players from moving through walls!** Unfortunately, affected players will still be moving too fast.