Of the places you listed:
Tokyo- Pretty easy to spend 6 days here, I feel that 6 days is a good amount to be able to see major things, but also have time for exploring.
Hakone- I did a 2 night stay here at a ryokan, it was very relaxing, when I went, (probably mid-week, I don't remember) it was almost deserted, and the onsen were wonderful to be in alone. A great place to wind down in after being in tokyo. You have to take the bus system to get here, which i found to be much, much more confusing than the train system (which is cake), so make sure you have the routes all planned out. I would imagine it only being so-so if its really crowded though.
Ogimachi / Shirakawago, Takayama- Never been to these areas, so I couldn't help on these. seeing as these are 1 day, they would probably be just day trips?
Kyoto- Definitely a great place all around for both old and new world Japan, I'm sure you've read all about the places here. While you are down there, and especially if you have a JR rail pass (below), you could easily pop down for a day trip to the hiroshima peace memorial- definitely a moving museum.
Nara / Mt. Koya- Staying at a Buddhist temple on mt. Koya was a defining experience in Japan for me. The most peaceful and serene place I have ever been in my life. The monks are wonderful people, and they will make your stay unforgettable. Getting up the mountain requires taking a crazy-steep cable car ride up the side of the mountain that pretty cool.
A JR Rail pass would probably save you a LOT of money on shinkansen. You have to buy a ticket outside of Japan at one of the authorized places, that you exchange at any major station once you get there. You could use a 14 day one, and exchange it a few days into being in tokyo, so it will last you the rest of the trip. Seeing as a shinkansen ride can easily be 10,000Y, if you ride the shinkansen 6-7 times it will pay for itself, not to mention you can use it on almost any local trains (but not the subways).
JAPAN RAIL PASS | ジャパン・レール・パス
If you have any specific questions about the areas I'll do my best to answer them, I spent a lot of time in most of them.