Mostly -
Tokyo to Nikko - Yes
Tokyo to Hakone - Kind of. The JR pas will get you to Odawara, then you'd have to use a different train company. You can get a Hakone free pass (annoyingly not free) which will cover the second part of your journey and unlimited use on certain buses cable cars etc in the Hakone area. I did like Hakone but had the advantage of a tour guide with us.
Hakone to Kyoto - I think you'll have to double back to Odawara then use your JR Pass to Kyoto.
Kyoto to Nara - yes (and as many have said, well worth it)
Kyoto to Miyajima via Himeji - yes
Miyajima to Hiroshima - yes -
Hiroshima to Kanazawa - yes - this will be the longest journey I think (and doesn't really go the route on the map from quickly looking - you'll be going back to Kyoto and changing there for Kanazawa
Kanazawa to Shirakwago - no, the only way is by bus and you have to reserve a seat
Shirakawago - Takayama - no, you'll need to get the bus
Takayama to Tokyo - Yes
Sorry if I've micromanaged that a bit. A quick note on Shirakawago - I did a day-trip and had mixed feelings about it. The village itself is lovely but the buses and buses of tourist take away from the atmosphere a little. My trip was during the winter though so it may be different during other times of the year.
For trains
http://www.hyperdia.com/en/ is the place to go. You can select only JR trains when searching timetables. Also while your Pass will be valid on most JR trains you can't use it on Nozomi, Mizuho or Hayabuse bullet trains. It's not a big deal, at most they shave 15/20 minutes of of a regular bullet train.
When is the best time to go? My wifes a teacher, so we are restricted to school holidays! I was thinking early to mid August for 2 weeks. Plane tickets seem cheaper then, however i expect there to be flight sales at the end of this month. I managed to pick up august 2017 flight to nyc for £500 each from virgin last january!
https://www.japan-guide.com/e/e2273.htmlPersonally my favourite time is winter but then I love the cold and snow. You can't really go wrong with spring or autumn TBH, I've done both but appreciate Autumn may be tricky for term times. I've never done the summer, I'd love to because there's a lot of cool stuff going on but my friends tell me the heat and humidity would kill me (I hate the heat)