For Christmas Lights, is it safe to run extension cord under garage door?
Yes to the last: you certainly need outdoor-rated extension cords for anything exiting the house.

Now, as to how to run the cord:  the comments cover your planned implementation.  I would strongly recommend, however that you or your friendly electrician run wire (properly inside conduit) from the existing internal outlet to a new external outlet box.  The hardware cost is trivial, and the work involved in attaching to the existing wiring, drilling a small hole in the wall next to the garage doors (or whatever is the preferred location) is easy.  Plus you now have an outdoor outlet for general use.