10/3 down to 12/2 for Air Conditioner?
You're always allowed to upsize wire

I don't know how your house came to have legacy air conditioning wiring that large, yes I do. Air conditioners have gotten more efficient and past owners may have retrofitted insulation.  

But regardless, if the requirement is for 14 AWG wire, you are always allowed to use the larger #12 or still larger #10.  

There is also nothing wrong with an unused white wire "coming along for the ride".  

However, when a white wire is used as a hot in a 240V cable, you are required to mark the white wire with tape.  I strongly prefer wrapping the tape around the wire; flags can easily be torn off, especially as they age. 

As far as breakers, if your panel is Eaton, Cutler-Hammer, BRyant, Challenger, or Westinghouse, Eaton breakers are almost certainly correct. If it's anything else, it's probably a hazardous mismatch, but anything is possible with Eaton; we'd need the panel brand and model, and the lettering on the breaker to be sure.