How can I patch a rotting wood door?
Dig out any rotted wood that you can get to.  Soak the wood with wood hardener around the patch area to stabilize the rotted/deteriorated wood that you couldn't get to.  Fill with bondo or wood filler in 1/4" layers until slightly bulging from the patch area.  Chisel/file/sand down when the final layer is almost completely dry with a sanding block.  Prime and paint.  The patch should outlast the rest of your door!