Stream a Video URL to Apple TV

I haven't tried it yet but according to these articles Ustream will enable you to do it:

• Ustream Launches iPad-Optimized iOS App, Adds Apple AirPlay Support
• UStream for iPad

No - that is not at all common.

Something is certainly causing the instability so you might need to isolate the potential causes one at a time.

• network issues (the most likely culprit but by far not the only one)
• Apple TV needing a restore or perhaps just a reboot.
• iOS device needing the same (restore or reboot).

Before restoring your iOS device, you might try the other steps first or also you could take your device to another network / Apple TV such as a retail store to test and make sure that's not the cause.