I guess I'll be controversial as well, and say that we did both Hakone and Takayama on our first Japan trip (which was 13 nights) and were very happy with both. Our final itinerary was as follows:

Tokyo 4 nights
Hakone 1 night
Takayama 2 nights
Nara 1 night
Koyasan 1 night
Kyoto 4 nights

For detailed descriptions you can see my first Japan trip report. In brief, we loved both Hakone and Takayama, which were very different from each other, and they gave us some sense of small town and countryside outside the major cities. I think that 1 night is plenty for Hakone, but would not do Takayama with less than two nights. We did Kanazawa on our second trip and also loved it, and didn't regret not doing them together on the same trip. Have a great trip!

We did Hakone as a day trip and Takayama (2 nights)and Kanazawa (3 nights) on our 3.5 week trip to Japan. I posted a TR if you are interested in details. We found all three places wonderful and worthwhile. Since you don't have the time to "do it all", I suggest that whatever you decide, you'll have a wonderful time. We are returning for our second trip to Japan in Nov.

<<Day 8/9/10/11 - Kyoto with a side trip to Nara (overnight?) and an evening in Osaka
Day 12/13 Back to Tokyo, flying home on the evening of day 13.
How does this sound?>>

You don't need to double back to Tokyo for an overnight on Day 12. You can just hop a shinkansen from Kyoto and then travel to Narita/Haneda all on the same day. The shinkansen average missing their schedule by 36 SECONDS per year. A shinkansen that misses its schedule by a minute is a shock. One that misses the schedule by an hour is unheard of.

I agree that traveling back to the airport from Kyoto on the day of your flight is easy. Just leave yourself plenty of time so you don't feel rushed. I sent my luggage ahead to the airport minus a small bag, and picked it up when I arrived. That would leave an extra night in Kyoto.

I agree to just head back to Narita from Kyoto on the day of departure. On our second trip, we traveled back from Matsumoto to Narita on the day of departure. This was more complicated than the Kyoto to Narita trip would be, and we had no problems. We also sent our luggage ahead to the airport which was super easy. You just need to do it a day or two before. Your hotel can help you to do this.

Hakone is easily one of the most beautiful areas in Japan. Fujigoko (Fuji five lakes)region just above Hakone is also beautiful and less populous. If you love lakes as your name suggests then I would recommend two nights in Hakone probably near Ashiko (Lake Ashi). BTW ko is Japanese for lake so every lakes name will end in ko. Just as yama is mountain thus Fujiyama. Anyway there is a lot to do and see in Hakone and for two days you'll just brush the surface. We will be going back to Hakone for four nights, our fifth visit, with rhkkmk next April and can't wait. See this site for some info on the area.

http://www.hakone.or.jp/en/

Aloha!

We stayed at the Fujiya hotel 2 nights back in 2002. Great historic hotel and easy access to sights.
We did the cable car/ropeway /pirate ship, then hiked along the old Tokaido road through cedars stopping at Amazake Chaya Inn for refreshments, then on to Hatajuku town for the wood crafts. Then took bus back to station,train to hotel, a great day.
http://www.japan-guide.com/e/e5205.html

Thank you so much everyone for your ideas and comments.... so appreciated and so much to think about!!!
I'll let you know what we decide!

Seeing Mt. Fuji with it's snowcap was not only the one single highlight of my first trip to Japan, but is also one of the top travel highlights of my life. Having said that, it took a return trip to Hakone near the end of my first trip, after the weather had greatly improved for me to be able to see Fuji-san.

My vote would be to keep your schedule flexible enough so that if weather is good while in Tokyo, you then make a day trip to see Fuj-san. A day trip will be busy but in good weather, will allow you to take the full transportation circuit (funicular, gondola and Lake Ashi boat cruise), see Fuji-san from multiple viewing sights, and visit the Owakudani (sp?) boiling valley. Granted, keeping this flexibility would likely preclude you from doing an overnight in Hakone but if your top preference is to see Mt. Fuji, you'll need to maintain this flexibility due to the variability of the weather. One other thing to consider--if you are traveling in October, it's quite possible Fuji will have no snow cap, which to me, would greatly reduce the classic view of it that you probably have in your mind.