This show is actually just an ad for Yamanashi. If you were expecting actual room activities, you were a fool.
The Fujisan World Heritage Center is of course a real place, in Fujiyoshida, Yamanashi. They’re all going to be in Yamanashi.
The myriad ‘local Fujis’ are mostly named after the area they’re in. The one exception this episode is Nanbukata Fuji, which is actually Mt. Iwate. One explanation is that ‘Nanbu’ (‘Southern Part’) refers to the side of Iwate that looks like Fuji (the pretty southern part; viewed from the north, Iwate’s kind of deformed). Another is that that side of Iwate looks like Fuji when Fuji is viewed from the south (i.e. Shizuoka). ‘Kata’ (‘partial’, ‘one (of two)’, ‘one-sided’, whatever else you want) perhaps refers to that the mountain’s deformed and only one side is pretty, or that only one side looks like Fuji, or that the south side is the side that looks like Fuji, or it looks like the south side of Fuji, or any number of things. It’s one of those things nobody actually knows. ‘Nanbukata Fuji’ might not be the best way to break up that combination (as it’s unclear where the adjectives end and how they attach), but it really doesn’t matter and it was better to be consistent with the other ones. Just for your interest.