There should be a drain hole in the bottom (as in underside) of the door!!
Has it been filled/repaired over?
That, and the missing plastic sheeting, might explain why it's pouring inside. It's supposed to go straight down and outside via the aperture between the door and the doorstep. No problem with having water inside the door, it's meant to do in and straight out again.
If you don't want to shell out for the proper plastic sheeting you can use duct tape over all the gaps and push the door card clips through it where you need to. Worked for me for a while.
Other possibility is it's rusted at the bottom and water's getting through the rust. Quite possible if you have no drain holes!