You can get some good savings by using a JR Pass, though you could increase your savings further if you wish (see below).
I presume you will start using your JR Pass immediately from Narita Airport - that is Day 1 - you just need to return back to Tokyo by the end of Day 7. The JR Pass covers all the places you plan to go to except the Fuji area. For that, you can take the bullet train to Odawara Stn., then to save money, buy a Hakone Free Pass. After seeing the area, return to catch the next bullet train to Tokyo. The Hakone Loop Course is a common way to see the area.
Regarding using the pass for local transport in Tokyo, you can use it on the Yamanote Line within Tokyo, and it does go to several popular tourist areas. It's unclear where you plan to spend your time however, and it's likely you will go somewhere the Line doesn't, so a 2 or 3 day subway pass might save you some money. You'd really have to ride your behind off for a 1 day pass to be worthwhile, however.
For where to stay in Kansai, you could stay at either Osaka, Kyoto or Nara, but to make things easier I suggest you stay near either Kyoto or Osaka Stn to make it faster for you. It's only about 30 minutes by train between Kyoto and Osaka. I have no idea on your budget so I can't really suggest any hotels - there are some for every budget though, so you could try hotels.com or booking.com or some other engine.
As I mentioned before about adding more value to your pass, what you could do is use one of the days at the start of your trip and instead of Tokyo, go take a day trip to Nikko. After seeing Kansai, if you still want to see Kamakura & Enoshima, you can use a Enoshima-Kamakura Freepass.
I would suggest rearranging your time in Kansai though. One day in Kyoto is barely getting your toe in the water. I suggest you add a 2nd day and then spend half a day in Osaka and the other half in the Nara Park area. The Todaiji Great Buddha is a must-see, but also be sure to feed the deer, see Kofukuji Temple and Kasuga Grand Shrine, plus two often overlooked gardens, Isuien and Yoshikien. After the temples of Kyoto close down around 5 PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are excellent.
For a Japanese inn (ryokan) with your own private onsen, the costs are extremely high. I have not stayed at any, but you can find some here, so perhaps there is one not too far from where you plan to go.
In your last days before leaving Japan, since you are going through Haneda, you might look at some of the subway and Haneda train combo tickets.
Best of luck.