It is really your call depending on the risk to the bird of escaping. I would definitely clip until you get the screen door fixed. All takes is, like, a one second window (pardon the pun) for them to escape forever. It happens all the time because people believe it just won't happen to their bird.
Clipping the wings won't make much of a difference if your bird is attacked by a dog. Dogs are faster. There are plenty of unclipped parrots who get killed by dogs ... so a cockatiel isn't going to stand a much better chance. I would suggest, instead, that you only let your 'tiel out under supervision and never, ever, leave it alone exposed with your dog.