I've done a bit of thinking (sorry for the delay), but I see this is going to be well in time for your trip.
Firstly, as you're planning on using the JRPass to do the Tokyo <=> Kyoto trip, you're already saving money, so you can view any other travel as a bonus.
You should plan on activating your pass on Day-3 to give you the full 7 days up until you are due to fly back home.
To cover Day 1-2, you can buy a combined one way narita express fare and metro charge card preloaded with ¥2000 at the airport as part of a deal for ¥3500. As the standard N'EX fare to Shinjuku is ¥3,110, you basically get underground travel for your first few days in Tokyo for free.
For day 3, I have a lot of love for naka-meguro/Daikanyama. It's full of cute little boutique shops relaxing bars and eateries in the evening. Really interesting areas to people watch, especially if you are already staying in Shinjuku (it's fairly close). The other highlight is the biggest fish market in the world at Tsukiji. You'll have to get up early to really see it in full flow (ideally be there before 7 - this may not be a problem depending on your jet lag), but you'll be rewarded with one of the most memorable experiences of your trip. You'll also be able to sample some of the freshest, and cheapest sushi/sashimi on your trip at the restaurants that cluster around the market.
You'll be going to Kyoto next, so I'd recommend you get as much experience of the modern side of Japan as you can!
For day 4 or 5, you should try and stay in Arashiyama. It's an area to the west of Kyoto, and is a really beautiful town, with some incredible temples and swaying bamboo groves. As Nara is so close, it would be a shame not to spend some of the day there.
Day 6-7 in and around Hakone sounds like a great idea. Try and stay at a Japanese Onsen in that area.
The curveball choice for day 6-7 is going to Takayama instead. Personally I'd prefer it to Hakone (which can be very touristy) as it's a little off the beaten track, but you'll have a great time in either place. You can get to Takayama in 3 hours by train from Kyoto(http://goo.gl/mi1oW).
Hope this helps!