Will it damage my screen / display if I leave it on all night as required by some third party apps?

No it won't, assuming you want to run a clock app or something like that. They are meant to run for hours at a time. It will wear the battery down faster if it isn't plugged in to charge, but I wouldn't worry about the screen. You will not be running the same app 24/7 so the image will be changing from time to time anyway.