Can I buy 10/2 with black and red instead of black and white?
I don't know, maybe not. You might not find it in a big box store, maybe you would have to get it from an electrical wholesaler. I haven't looked for it.
My wife already picked up the 10/2 (black, red, copper) I wanted the 10/3.
You mean black, white, copper?
So we used heatshrink tubing for the white. The inspector really loved that and felt that's how everyone should do that, so we earned some brownie points there!
You put red heatshrink on the white wire? That's fine. You could also use a wrapping of red electrical tape.
That's what I wanted, the 10/3 w/ ground. I could have used all those wires, couldn't I? Would it be wrong to have one of the neutrals going to ground the metal conduit box and the other ground wire going to the receptacle?
Here I am confused. Neutral (white) is not ground (bare copper). The neutral wire has white insulation on it to indicate it is a live (current carrying) conductor, and must never be allowed to touch bare metal parts.
What I did, because I only had the 10/2 was run a pigtail off grounding screw on the receptacle, run a pigtail off the grounding screw on the box, and then twisted them together and used a wirenut to hold them together.
Again, confused. You had 10/2 cable with black (L1), white (marked red, used as L2), bare copper (no insulation, ground wire). The bare copper wire shall be connected to the ground screw on the receptacle, the box, and any other metal chassis parts. Pigtails may be used for this as necessary. Under no circumstances shall a bare copper (ground) wire be connected to a white (live) wire.
I do not understand why you say "because I only had the 10/2" ? Your 10/2 has a bare copper ground wire to connect to ground screws. Where was the problem?