Copper is worth around 40-50% more than brass, so separate it if you have enough quantity to make it worthwhile. If it's only 4 inches I wouldn't bother. However, if you plan to do this very much, keep a separate pail for copper.
Bend it back and forth until it cracks off. Or, use a hacksaw.
I've found that some bathroon faucets have a chrome shell over the insides. There is just alittle brass and copper. I just throw away the shell and throw the insides in my brass barrel.