It looks like the door has been leaking for a while due to the rot of the subfloor. The only thing I can think is that the driving rain is getting in the door track and then getting under the metal, hitting the subfloor where the screw holes are?
I've this exact same problum a couple hundred times.
The tracks pluged up, I'd bet money there's no or the wrong flashing under the door, and most of all there should never be any soild suface (stoop, deck, patio) any closer then 4" under a threshold.
Have it even and it's going to leak into the house 100% of the time and no mound of caulkings going to stop it. http://www.bing.com/images/search?q=...2B6F63&first=1