What is the standard height of an electrical outlet?

There isn't one.

In an existing house it's the same as all the others.

I'd say 12"-18" is average.

18" to the center is common, but it is more accurate to mark at 20" and have top of 4" box at the 20" mark.
If remodeling, maintain the same as existing,
As well as horizontal/verticle.

There is no standard height for receptacles in a dwelling. 16" from floor to top of rough in box is normal in new construction. Receptacles above countertops are generally placed at 48" from floor to top of box to avoid backsplashes. Note that code requires receptacles on regular wallspace to be not more than 66" from floor and receptacles above countertops to be not more than 20" from countertop to meet wall space requirements.

16" from floor to top of rough in box is normal in new construction. Receptacles above countertops are generally placed at 48" from floor to top of box to avoid backsplashes. While these are the same numbers I use, I would not nearly call them "normal".

Ask 20 professional electricians this same questions and you'll get 15 different answers.

According to code, I believe it's about a foot from the floor to the bottom of the box. I may be wrong, though.

According to code, I believe it's about a foot from the floor to the bottom of the box. I may be wrong, though.
The code has already been stated. Keep in mind this is for required receptacles ONLY. Once code minimum has been met you can put them where ever your heart desires, including the ceiling.

Here:

Note that code requires receptacles on regular wallspace to be not more than 66" from floor and receptacles above countertops to be not more than 20" from countertop to meet wall space requirements.