6) Was there any eccentric experience that you would consider so Japanese we should do no matter what?
I don't know if I would say eccentric and I'm honestly not sure who this is targeted at because it's a PIA to get to but....
The number one weirdest, most "eccentric", and possibly offensive, depending on your religious bent, experience I have ever had in Japan is visiting Jesus' "Grave" in Shingo, Aomori Prefecture. It's in the middle of nowhere, and you're not going to get there easily without a rental car because buses are just not that frequent. Once there, you will find out that Jesus visited Japan as a young man before returning to do all his New Testament preaching. You will also learn that his family name was "Christ" and that he had a twin brother no one's ever heard of before name Isukiri "who casually took Jesus' spot on the cross." (I am 99% sure that is an exact quote from the literature available there in English!) Jesus then returned to Japan where he became a rice farmer, got married, had a bunch of kids, and eventually died at the age of 106. Isukiri's ear also has a grave at the site. There are tons of "proofs" that this story is "true." And it sort of has to be seen to be believed. They also sell Dracula (garlic) ice cream there, and I bought Dracula the Cool (aka garlic mint, you don't taste the garlic you do taste the mint.)
https://www.japan-guide.com/e/e3781.html
The area is very rural and pretty. We got caught in a blizzard on our way to Towadako but that is neither here nor there. If you have a car in Japan and are not offended by its existence, it definitely qualifies as weird and eccentric.
I collect fashion dolls, so it's possible to non-fashion doll collectors Licca Castle would count as a weird and eccentric place. You can get there via public transport (shinkansen to Koriyama, local train to Ononiimachi) http://www.liccacastle.co.jp/ (And yes, it is a small Western style Castle in the middle of nowhere Japan.)
I also collect Asian Ball Joint Dolls. One of the big makers of them, is the Japanese company Volks. They have shops throughout Japan, but a special store/garden/cafe in Kyoto in Arashiyama called Tenshi no Sato. You must be over 12 and have a reservation to visit. The garden is lovely in season and quite quiet. Doll owners can be a little weird (and I say this as a doll owner) The Omukae ceremony is pretty cracked out and (keeping a straight face during it is really hard) there is a statue of the Virgin Mary Doll Mother surrounded by little doll angels as well.
http://www.superdollfie.net/tenshinosato/about_en/
For "eccentric" places I've not been:
I would love to visit: http://www.muvalley.com/ (Mu Valley)
It looks pretty insane, because made up civilization. It's definitely meant for the Japanese since they don't have an English website. Again, it's only accessible via car.
Another semi eccentric place and going along with the question regarding toilets is:
https://www.japan-guide.com/e/e4883.html The Toto Museum! And unlike Mu Valley is actually accessible via public transport. It's right near Kokura which is a shinkansen stop.
I guess I would also probably include Spa Resort Hawaiians in Iwaki
https://www.japan-guide.com/e/e7777.html
It's a Hawaiian Theme Park in Iwaki.
I like onsen and I think given the chance most people should give them a try. If you have a tattoo that might be harder, but there are some tattoo friendly places now. But for more "unusual" onsen, there is: Yunessun in Hakone https://www.yunessun.com/en/
A bathing suit onsen with interestingly themed onsen baths including: sake, red wine, coffee, and green tea. I've not been since they've sold off some of their land to build a new hotel, but they also use to have a limited themed onsen depending on season. The weirdest I ever saw was Keroro Gunsou: https://en.wikipedia.org/wiki/Sgt._Frog which I swear looked like it consisted of Palmolive dish soap.
Not to be outdone Osaka has SpaWorld http://www.spaworld.co.jp/english/ which has themed onsen and is more traditional in that sense that you enjoy said baths naked.
I personally liked Zao onsen more along with Hirayu, but Yunessun and SpaWorld are definitely more on the eccentric side.
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