Tokyo-Hakone-Kyoto-Osaka

Hi there
I am looking for some tips on my upcoming trip to Japan and also to understand if it's worth getting the Japan Railcard for my partner and myself.

Overall we are there for 13 days:

• Day 1 (Monday): arrival midday in Tokyo Narita.
• Plan to stay in Tokyo until Saturday (Day 6) but with a day trip possibly to Nagano to see the snow monkeys (is that possible in 1 day?)
• Day 6 (Saturday): train to Hakone until Sunday
• Day 7 (Sunday) or Day 8 (early Monday) train to Kyoto
• Day 11 (Thursday): train to Osaka
• Day 12: day trip from Osaka to Hiroshima
• Day 13: Osaka with late flight

1) does this itinerary sound feasible or are we too ambitious/not enough
2) will the JR make sense?
3) will we need other type of transport tickets there?
4) if JR does not make sense, how do I get train tickets in between?
5) Would you suggest a different order (starting in Tokyo and ending in Osaka)
6) Is there anything else you would suggest doing (wanting to sample the cities like Tokyo, historical sites like Kyoto, onsen, a bit of wildlife, lots of culture, some rural experience and of course lots of food. I've also heard it's possible to visit some wineries near Tokyo - do you know any/can you recommend?

Many thanks for all the help

Shyamal