when and which pass to take for our trip

Hi there,

Sorry for the delay in response, I've been zooming around Japan on the bullet train myself for the last few days.

Machuoto = Matsumoto? In general, I do find it hard to match a pass to your travels, this due to the fact that you are having longer breaks in between travel.

That said, I think your best option would be a 21-day JR Pass for:
0 to 5/10 Aomori
5/10 to 9/10 Machuoto
9/10 to 12/10 osako
12/10 to 15/10 tanabe
15/10 to 20/10 kyoto
20/10 to Atami

and then lastly, normal tickets back to Tokyo. Also because the JR Pass will give you a lot of extra flexibility, and it would be easy to add day trip, such as a day trip from Osaka or Kyoto to Hiroshima.

As for the hotel, I think it will be okay but just give them a call to confirm.

Kind regards,
Daniel