Q: Can I share video in a photo stream?

Not at the present time, however Apple has announced that there will be an ability to do this with iOS 7.  From http://www.apple.com/ios/ios7/features/:

iCloud Photo Sharing.

With iCloud, you can share your favorite moments with your favorite people by creating a shared photo stream. They can post photos, videos, and comments to your stream, and it all appears on everyone’s iOS devices — automatically. The new Activity view lets you see the latest updates from all your shared streams in one place.

new video share feature doesn't work. on the other device people only see that video is shared and it keeps loading forever

Haven't tried it yet myself.  The servers might just be overloaded right now.

Even after 1 week, my movies still don t play in the stream. When I click on them I only see the first frame and a circle turning. I found nowhere a note from Apple that this is a known bug.