Electrical – Running a wire from an existing interior outlet to a new exterior outlet which is lower than the interior one

You have a fairly simple project. Assuming you intend to tap the interior outlet for power, here are the steps and materials:

1. First, check that you have enough room in the interior outlet box to introduce one new piece of 14/2 or 12/2 NM, whichever is the same size as in the box now. I will assume you know how to make a parallel electrical junction in the interior box.
2. Next, run that short piece of wire through a clamp in that box, across/through the wall cavity and out of a 1/2 to 3/4 inch hole in the exterior wall close by. I don't know what kind of siding you have, so I can't tell you exactly how to make the hole, but a spade type wood bit works well on many types of material. Minor fishing of the cable will be required, but not difficult, especially if the interior box can be removed partially to improve visibility.
3. Once you have fished the wire from inside the interior box to the exterior, made your interior electrical connections and secured the cable inside, caulk the exterior hole around the wire with silicon.
4. Feed the wire into an exterior surface mounted box using the proper grommeted fitting. Caulk around the back edges of the exterior box with silicone (to make the back water tight) and fasten it to the wall.
5. Now you can connect your new GFIC outlet and install your weatherproof cover.

Of course, use proper safety procedures: turn off power at breaker panel before working in the interior box and doing any wire connections. BTW, if you have vinyl siding, there are special exterior boxes that will fit nicely. These are available anywhere. I know this is a long answer and maybe a bit confusing, but it is really a simple task. Good Luck

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.