I'm trying to replace a kitchen faucet. The stud and nut holding the faucet down are both turning. I'm out of ideas.
WD-40, A box wrench and a pair of pliers. Apply WD-40. Get the wrench on the nut and gently hold the bolt down as low as it can go. Turn your wrench while holding the bolt. Hold the bolt with the pliers harder until it frees up. Be careful to not marre the threads because you will have to bring the nut down through the damaged section if damage occurs.
A better way is to cut a slot in the bolt with a cut off disk on a grinder and use a slotted screw driver to hold the bolt in place. This method is potentially dangerous and not recommended if you are not comfortable and experienced with a grinder.
What you show is a stud coming through a plate washer with a nut.
If you continue undoing the nut then the stud should come out of the faucet body but it may be in a long way.
If the thread is damaged inside the faucet and it is just rotating then you may be in for the long haul with a hacksaw or drilling the nut until it can be split.