Can I use 10 gauge wire off of a 20 amp breaker for outlets, lights or a window unit? I am wiring a barn and have a lot of 10/3 wire I would like use up.
Yes, you can use 10 AWG copper conductors with a 20 ampere breaker. The smallest size conductors you could use with a 20 ampere breaker, are 12 AWG copper conductors. There's no problem using larger conductors, other than cost to you, and difficulties associated with working with thicker conductors.
You're always allowed to upsize wires, although that doesn't mean terminating them will be easy. Breakers can always accept at least 1 upsize, because they accept aluminum wire (which itself requires 1 upsize).
Most receps can handle #10 on the side screws, just check their instructions (you have to look on the web) or their labeling. Also try screw-and-clamp type receps, and if that fails, aluminum rated receps (CO-ALR) are a sure bet because a 20A circuit requires #10 aluminum.
I gather you want to ignore the red wire? Consider not doing that, and putting the red wire to good use as a multi-wire branch circuit. However, you do need GFCI in this decade, and GFCI is tricky with MWBCs. You can't share LINE and LOAD neutral on the same wire. It comes down to strategically chosen $17 GFCI receps, or a $80 240V GFCI breaker.
A great example is if you want to dedicate a circuit to a 120V saw and 120V dust collector. Putting them on the same circuit is usually a mistake. So punch them down to a 2-pole 20A breaker, then feed two 20A GFCI recepts, one off red, the other off black, splitting the neutral on the LINE side (you wouldn't use LOAD at all).
If you wanted to feed additional outlets off the LOAD terminals, you must do it with /2 cable only (or don't use the red at all). This cable must attach all its wires to LOAD, it cannot interact with LINE (which means it can't share neutral beyond the GFCI recep).
Lastly, keep in mind that when you're working with #10 solid wire, it's less like wire and more like bus-bar. Use LARGE junction boxes (4-11/16 square with a 1-gang mud ring is not excessive here) with lots of room for the wires to lay down.