Electrical – New wire to the old receptacle, using the receptacle like junction box and outlet box

Read inside the box. If you install 14 AWG conductors to the new box, instead of 12 AWG conductors. You'd need at least a 22 cu.in. Box, or a 3" x 2" x 3 1/2" device box.

• 8 for current carrying conductors.
• 1 for grounding conductors.
• 2 for device fill.

11 total, times 2.00 cu.in. for 14 AWG conductors = 22 cu.in.

However, since you're using 12 AWG, the calculation looks like this...

• 6 for 14 AWG conductors x 2.00 cu.in. (12 cu.in).
• 2 for 12 AWG conductors x 2.25 cu.in. (4.5 cu.in).
• 1 for grounding conductors x 2.25 cu. in. (2.25 cu.in.).
• 2 for device fill x 2.25 cu.in. (4.5 cu.in.).

Total = 23.25 cu.in.

For more information on box fill, check out this answer

As for connecting the new wire. There should be no problem attaching it to the receptacle. However, since you're using 12 AWG conductors, you'll have to use the screw terminals.

If you want to do pigtails, you can. I know some guys don't like the backstab connections, so if you want to avoid using them you can do pigtails. Ideal 76B® Red Wire-nut® Wire Connectors are rated for 2 to 5 #14 conductors, but check with the manufacturer of the connectors you're using. Again, if you're going to do this, you'll want to use #14 conductors instead of the #12 you've installed.

Electricity doesn't care much whether it runs vertically or horizontally (it really only cares how far it runs).

Often the easiest way to add an outlet or switch is to run it down through a basement or up through an attic. So long as the cable is properly buried in the wall cavities and you don't unreasonably compromise structural members, there should be no problem.

This assumes, of course, that you are not overloading a circuit, that you are using the proper gauge of wire, and that any exposed cabling is in a permitted area (usually an unfinished basement or attic is okay under most local codes). You also need to make sure cables are properly tacked down and not subject to damage, and that as you drill, you are not hitting other cables, pipes, or leaving hidden cables too close to a surface where they might get damaged in the future.