Before you unclip your daisy from the anchor always load the rappel and look at the rope going through your device. You can even pull yourself up the rope a bit and lower down while watching the rope slide through.
If you are backing off something where you have to try and save gear the heavier person goes first. The anchor will be one piece that's loaded and a backup piece that's unloaded. The lighter person watches the loaded piece as the heavier person rappels. Then the lighter person pulls the backup and rappels off the single piece.
This is especially prevalent with ice climbing where you are going off Abakalovs. You place one Abakalov and a backup screw, again the lightest person watches the Abakalov as the heavier raps, then he removes the screw and goes off the single Abakalov.
A couple years back I was rapping off an ice climb with some new gloves. The rope was a little bit icy and I got about 1/2 way down this free hanging rappel and hit a patch of ice on the rope and it started slipping through my glove. I sort of freaked out then whipped the glove off and grabbed the rope with my bare hand. Ever since then I've used a prussik on the brake side.
Just last week a climber died on rappel at Beacon Rock OR. Apparently he didn't clip both loops of rope when he clipped his device.