I want to run an alpha test for my game but don't really want to go through the hassle of using steam. We will be using steam for beta testing, but for alpha testing is it possible to upload a game to itch.io and then after a certain period of time delete it?

If the game is set to "no payments", you should be able to delete it easily when you're done. But we hope you'll publish the finished game on Itch after that. ;)