As I understand it my patio doors were supposed to be installed on a
step down cut into the slab so it wouldn't leak water into the house.
They weren't and I have four of them. Whatever the reason they all
leak.

This sketch shows one way they leak. This leak doesn't seem to bother
the areas if adjacent tile but ruins an area of adjacent carpet.
http://img101.imageshack.us/img101/8...odoortrack.gif

This picture shows the other way they leak. This leak rots both door
jams on all four doors.
http://img412.imageshack.us/img412/3...ckp8064414.jpg

First, does anyone have a way to fix this without removing the doors?

Second, my idea is to seal the place it's coming under the bottom
plate (sketch one above) with something that will bond to the aluminum
and the cement, is permanent and is impervious to water.

What product do you recommend?

Then to seal the place where it leaks in picture two with something (I
assume the same stuff) that will bond to the aluminum, is permanent
and is impervious to water.

What product do you recommend?