Great question — answering as the UphillTech team:
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.
Full answer by Carl Witthoft: https://diy.stackexchange.com/a/102608 (CC BY-SA 3.0).
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