Does the video have a cloud symbol next to it? If so, it's not on the iPad. It's on iTunes, and the Video app is showing you that you can download it again by tapping on the cloud-with-down-arrow symbol (shown on the details page).
Just tapping on the icon will stream the video instead of downloading it.
The reason there is no Edit button is because there is nothing to delete. When you have at least one video downloaded the Edit button should appear.
If you don't want to see your previously purchased videos in the Videos app go to Settings > Videos and turn off Show All Videos.
Anyway, that's how it should work. I've got iTunes Match turned on, which seems to have locked Show All on in both the Music and Videos app; which makes sense for Music, but seems odd for Videos.