What is the minimum distance an electrical outlet can be from an exterior door?
There is no minimum distance specified in the NEC. It can be as close as is physically possible given the constraints of framing, trim, etc. There's several rules about maximum distance and spacing and when to include door swings or not, and what constitutes a usable wall requiring outlets in the first place. But no minimum.

Now, conceivably, someone could interpret that since the outlet is within 6 feet of a "damp" area (patio) that it may need to be GFCI protected. I personally believe this stretching the intent of the NEC too far since the theoretical appliance in a damp area would have it's cord running through an open exterior door into a different room.

There could be other requirements by Washington state or your local jurisdiction. The NEC is mute on this matter, but you should inquire with your local building authority to be absolutely sure there are no other applicable requirements or "creative" interpretations of GFCI requirements.