It's a freezer. As explained in the second video a previous tenant got the landlord a deal on a glass-front fridge that didn't have a freezer. So someone put small freezer in that closet. He moved the freezer downstairs and converted the closet.
No idea. Maybe it was originally like a linen closet?
Separately, and why I posted it: More and more I'm thinking I need to get a 3D printer even though I don't really have the space for it, certainly don't have the time for it, and don't want to spend the money on it.