I love the idea of time trials but I think the time challenges should give you unlimited time to complete the level so you can learn the level without getting booted all the time and then when you finish it could let you know if you'd passed fast enough to get the completion reward.
Also time challenge would be better if it was simply available for any completed level without having to start from the beginning and do them in order. This would mean if I'm stuck on one particular level I could still attempt other ones as long as I'd already completed the level normally.