Some hotels have monthly rentals, and long-term residents. But nobody can actually buy a room in a hotel; they'd have to buy the whole hotel!
Yes, sometimes it's possible. I've seen hotel rooms offered in Munich. The price was more or less the same as the price for any other apartment you'd buy. The difference was that there was a long-term lease contract with the hotel, which would pay you rent. I found the deal pretty one-sided, because you'd get a monthly rent of, say, EUR 400, and the hotel would then rent out the room for EUR 80 per night. Also, it was not possible to stay in that room yourself, even for a few days. So what's the benefit of owning that room at all? I didn't see any. The interest yield was a less than 5%, because of the low rent.
Yes, you can. You need to contact the hotel and see if they're willing to do that - the only problem is that in the contract you don't own the land or property, you simply get a 60-80 year lease.
No because then it would not be a hotel, it would be an apartment building....