No idea why it wasn't mentioned earlier but an important part of Hakone is currently closed to visitors.
Owakudani is out of bounds and so is a critical trajectory of the ropeway. This makes the trip to Ubako (where the Green Plaza is) into a long circuitous bus trip rather than the short, exciting rope way excursion with marvellous views that previous visitors will have enjoyed. If you do decide to go to the Green Plaza remember: males and females are segregated for bathing (females get the best view of Fuji San), be aware of the times set aside for hotel visitors vs guests, check to ensure that the onsen are operating as standard and not undergoing cleaning/maintenance on the day of your visit.
Without knowing your hotel in Tokyo nobody is going to give you the absolute best solution for getting there. Figure out where you're going and feel free to ask again.
As a "must see" with a 12 year old I would recommend Nara. I imagine that the Studio Ghibli Museum is already in your list. If you want to stay in the kind of farmhouse featured in Tottoro, there's Hattoji villa in Okayama
http://www.international-villa.or.jp...illa-line.html
Less of a commitment is a wonderful collection of farm houses, the Nihon Minka En, in the outskirts of Tokyo. This is one of my very favourite places in Japan. My own daughter also loves it here.
As you are looking to minimise the pain of hauling luggage across the country, rather than stumping up for the green car, consider using the takkyubin service instead.
For a modest amount, your hotel in Tokyo will send luggage to your hotel in Kyoto/Osaka or even directly to the airport (your Kyoto hotel will also happily send on luggage too). If you send the bulk of your luggage the evening before you leave Tokyo, it will be waiting for you in Kyoto. Your hotel concierge will be intimately familiar with the details and can arrange everything for you.
This service (and its economic price) is one of the main reasons you generally don't see a great deal of suitcases in the bullet train luggage racks.