The 7 day JR Pass is 28,300 (USD price depends on the current exchange rate) which will cover all your travel on the JR from Nov 4th until Nov 10th. This would include all of the above travel except for the area around Fuji (including Hakone) but even there savings are to be had. Let me know if you have any specific place you wish to visit and I can give you detailed route information.
Going on to your other
Arrival at Haneda Airport
I do think that its a good idea to stay at Haneda airport and travel on this next morning, this way you can save money on accommodation costs. The first train connection to leave 5:17 am here is the link in Hyperdia, look at Route 2. It is still a good journey but at least you can relax once you arrive at your friends place. You can activate the JR Pass later on at one of the main station in the Tokyo area, so there's no need to wait for the ticket office to open.
Tokyo and surrounding Area
For the other days in Tokyo / Ota, I would stick with normal tickets or buy day passes because travel around the metropolis of Tokyo is relatively low priced, this is also why I would recommend a 7 day JR Pass instead of a 14 day pass.
On to Kyoto
I would use the JR Pass to travel to Kyoto as Shinkansen travel can be rather expansive, you can also make some great day trips from Kyoto as mentioned before, all these are covered by the JR Pass. There are not that many trains in Kyoto itself and the best way to get around is by bus. A 500yen day ticket is available which allows for unlimited travel on the day of purchase.
Back to Tokyo and flight back home
You can use the JR Pass to travel back to Tokyo with the bullet train, this trip is completely covered by the JR Pass. I would recommend buying normal tickets for the last couple of days in Tokyo.
I hope this helps!