If a wire is rated 10A, 120v AC. How many amps could I put through it of 13.8v DC?
Unless you're working with RF (e.g. high-frequencies, > ~100 Khz), or really, really large wire (cross sections in inches), Amperes are Amperes are Amperes.

As such, a wire rated to handle 10 amps can handle 10 amps, independent of the voltage.

The voltage rating of wire generally relates to the breakdown voltage of the insulation. Basically, with wire rated to 120V, you can be confident that normal handling of the wire while it's energized with 120V will be safe.
If you were to put ~1000V on it, you might encounter issues with the insulation breaking down, and it could possibly shock or electrocute someone. However, this is a safety issue, and does not affect the wires ability to carry current.