I'm glad this happened. If it didn't, I may not have searched for a YouTube alternative in the App Store. I highly recommend TubeBox, it's brilliant. Better than the YouTube app ever was, and it lets you repeat them and download them. (You can't save them off the device though I don't think.)
I'm pretty sure it does.I use my fourth generation iPod touch and output video from the YouTube app with an Apple composite AV cable. I don't think that'd work with the mobile site.