I think this happens every year, but why is that? Someone will surely upload them anyway elsewhere so why wouldn't CBS keep them up to get the ad revenue/views or whatever?
You can watch on the Tony Awards website - plus lots of other great past performances....
No you can't. They simply provide you links to iTunes and Amazon to purchase performances so you can view them. The Tony Awards website doesn't have any of the Tony Awards performances posted to view for free.
If you google "Full Tony Awards" you should be able to find a full video of the Tony awards on websites.... that's what I did. It's not the individual performances so if you want to see a specific one you're going to have to skip around a lot but it's better than nothing.