Need a new outlet

Not an electrician..

To minimize damage to the exterior stucco I would drill a hole through the wall (from inside), adjacent the interior box (same side of stud) with a long bit. Pull the short wire and then mount an outside electric box on the stucco surface. No need to cut out the stucco, unless you want the box recessed.

Is the garage box on GFCI?

Sure. That's how the boxes in your house are connected (daisy-chained).

Don't know about "code" (we don't pay too much attention to that in Houston) but it would seem even better than lower down - less chance of water infiltration, weeds, lizards, etc.

Prudence indicates a GCFI outlet as well as a weatherproof box.

That should be fine. You should be able to determine by snaking into the wall from inside the existing outlet, just how far up or down you can go before hitting any cat beams. If the outlet in the garage was installed within the last 25 years or so, it should be GFCI protected already, if so, no additional GFCI protection will be required on the outside. The current NEC does require a "while in use" cover for outside outlets

If you have space within the wall cavity, and want to avoid a bulbous in-use cover that sticks out of the side of the building, I suggest a retrofit version of the Arlington InBox:

I like to have a few outlets around that height. They're a lot easier to get to. At standard outlet height, they're harder to reach and tend to get hidden behind things.

BTW, I know someone else who put an outdoor outlet up high. It's intended use was for holiday lights on the roof, so a low outlet would require a longer extension cord, which would get in the way more.

Who is Prudence, the building inspector?

How about replying to the person who said that?

Maybe it is me and I am not reading this correctly.

I would cut a 4-6" wide path on the inside into the drywall until I got to where I wanted the box. If using a surface mount I would then drill a single wire sized hole. to allow the wire to enter the back of the box. If using the retro then you would need to cut a hole the size of the box.

I would go with a surface mount bell box as they are easier to seal around and keep the water out of your house.

The drywall is a lot easier to repair than any damage you do to your stucco or substrate. The path only needs to be wide enough for you to insert your drill to keep the wire in the center of the studs. Please don't do something stupid like notching the studs and putting the wire in that. It will bite you in the butt in the future.

Also make sure it is GFIC protected. Your garage should already be that if recent contraction.