You could try pulling the door together with a pipe clamp.
When you have it tightened up as far as it will go put some long screws
in from the edge of the door to connect the stile to the bottom rail.
You can usually find up to about 5" screws in the specialty screw department
at the hardware store.
Drill about a 3/8" hole in from the edge about 1/2 way through the stile. Then
run the screws all the way in so that they have about half or more of their
length into the bottom rail of door. If the door is a hardwood you will probably
have to drill a pilot hole for the threaded part of the screw.
You can plug and sand the holes in edge of door or fill them with wood filler.
I like Bondo for paint grade work but don't let it get away from you as it hardens!
Regarding sealing; I use a waterproof glue to seal the bottoms of my exterior
doors. I usually thin it down a little so that it brushes on easily. Put an extra
coat on the end grain area for added protection.
If you have a door shoe on the bottom of your door; you can seal across
the top of it with a polyurethane caulk. Put a piece of blue tape on the door
just above the shoe and at the top of the door shoe. Apply bead of caulk
and press in and wipe off the excess with your finger. Pull off the two pieces
of tape at the same time while caulk is still wet. Messy little job but effective
in keeping water from slipping in behind the door shoe.
I hope that helps!
Richard The Door Guy