What's this piece under the nut on my faucet, and could it be stopping me from loosening the nut?
That piece is a threaded sleeve nut that is actually holding the faucet in place. the nut at the top just holds the bracket in place while the sleeve nut is tightened or loosened with a large slotted screwdriver. Support the bracket with an adjustable wrench while turning the sleeve nut counterclockwise with slotted screwdriver to remove the faucet.