Electrical – I’m trying to install a wall outlet where 3 Romex cables meet and are twisted together. Which goes where

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.

There are multiple combinations possible.

From what you have said so far here is what I am thinking:

Cable A or B is the hot feed from the main panel. The other cable, either A or B is the feed to the other receptacle. Cable C is the feed to the switches and the red wire is the switched hot wire returning from one of the switches.

It sounds like at least one receptacle in the room was switched for a floor lamp. It doesn't have to be split wired both plugs on the receptacle could be switched.

It could be the receptacle in your drawing or the other or both. The switched hot should be the red wire. If you connect the red wire to a receptacle you should not connect a black wire. Unless you are going to remove the connecting tab on the hot side of the receptacle and split wire the receptacle.

You should take some voltage readings to determine which cable is the hot feed from the panel. Then, with the circuit shut off, you can use the resistance or continuity setting of the meter to determine which cable goes where and which wire is which.

Good luck!