When we were being squired around Chiba in our friend’s car, my blue badge from the US was accepted and we would be pointed to the designated spots, if they weren’t immediately obvious.

Yes to both.

I was able to park my scooter right outside the shops, alongside peoples bicycles. Never had an issues.

I’m from California and used my blue parking badge permit as “proof” for the handful of discounts I got. I will admit I had no idea I could do that until midway through the trip, when an employee at the Toji Temple gate suggested it and asked if I had some sort of verification. I expect that should be fine to use for other places.

As far as I see it on the website, it looks fairly accessible. They have wheelchair rental service, too.
https://kagoshima-barrierfree.com/barrier-furee/%E7%9F%A5%E8%A6%A7%E7%89%B9%E6%94%BB%E5%B9%B3%E5%92%8C%E4%BC%9A%E9%A4%A8/