In this video a hotel in Belarus is shown with rooms for $4.10 per night. But the city is not mentioned. So, is this the cheapest hotel in Europe and where is it located?
No, it is not the cheapest hotel in Europe.
There is no clear definition of the difference between e.g. hotels, hostels and bed & breakfasts, but it is not difficult to find cheaper hotel-like lodging on booking.com in other European countries. I just tried a search in Ukraine for an arbitrary date in October and found about 60 offers for €4 per night or less.
I would also assume that many of the cheaper offers are not even advertising on the better-known booking sites, since it is too expensive. In Ukraine, there is e.g. very cheap lodging available on most train and larger bus stations, but you will not be able to book these online.
The hotel shown in the video is Zhilkommunhoza in Krasnapolle. That was already answered in comments to the YouTube video.