They're not fake since they have hinges, but the wall they're against looks to be exterior considering the placement of the window, so they might have an actual vent and some extra storage around the outlet.
Maybe but I've seen a lot of real cabinets above these too. The house I grew up in was like this and my GF had one in the house she was renting too. Seems very common unfortunately.
This is what I have almost always had in my place. Almost identical range hood with a cupboard around the vent tubing to give some storage for spices and shit. I assumed that's how it was in the picture cause that's literally all I've known.
I would advise against keeping spices there. Heat and humidity are bad for them. Or store most of your spices somewhere else and only small amounts near the stove.
I keep my tamale steamer/giant stock pot in there because it fits nowhere else and I don't use it often. Also the coffee pot from my old coffee maker, as insurance against breaking the one that came with my new coffee maker. It's an awkward space, perfect for awkward items.
Fake hoods have a vent on the front to helpfully blow the steam and fumes back into your face. This doesn’t appear to have that, so it’s probably real.
How did you know the hood is fake? Mine has a cabinet above it too but the hood duct is run through the cabinet and boxed in leaving room around it to still store things