Can I still install iOS 7 beta 1 on my iPhone 5 and update over the air?

The title says it. Does it also fix the need to not have a registered UDID?

Updating to any beta version of anything is a bad idea. Just wait a couple months and youll have the fully functional, perfectly safe ios 7.

Yes, though it's not recommended. You should do a clean install for each new beta.

Yes you can! Here's a video showing you the steps

You still need to register your UDID at a site like http://www.instantudid.com/ or your device could get bricked by Apple.