JR pass worth it?

I am traveling to Japan for 11 days and will be traveling outside of Tokyo for 5 days. I want to know if a 7 day JR pass would be worth it for this trip. I have heard that the JR pass cannot be used to go to Hakone, is this true? If so, then would it be worth it to just be using the JR pass from Tokyo to Osaka, Osaka to Naoshima, and then back to Tokyo? Also, can the JR pass be used for a day trip to Kamakura?

My itinerary would be:

22-25 - Tokyo
25 - Day trip to Kamakura from Tokyo
26 - Tokyo to Hakone, spend one night there
27 - Hakone to Osaka
28 - Day trip to Kyoto/Nara
29 - Osaka to Naoshima
30 - Naoshima to Tokyo

Use the 7-day pass to get to Odawara and then the other rail company (non-JR) to and around the Hakone area. Odakyu is the company that runs Odawara to Hakone-Yumoto and they are not JR.

The 7-day pass makes sense for your schedule.

You can get JR trains to Kamakura.

JR pass will get you to odawara which is the base of Hakone area. Then you can get a hakone free pass to cover travels around hakone area.

Yes to all other questions.

You already asked this.
Yes, the JR Pass would pay off for your plans. For Hakone though, it can take you up to Odawara, and then you can buy a Hakone Free Pass.
http://www.japan-guide.com/e/e2358_008.html
http://www.japan-guide.com/e/e5200.html
The Loop Course is a common way to see all the sights and takes a full day.
http://www.japan-guide.com/e/e5210.html
On your return, you can save some time by returning to Mishima Stn, then catch the bullet train going to Kansai.
And yes, you can use the pass to go to Kamakura. You'd have to pay for the Enoden train if you wanted to also add on seeing Enoshima.
You can use the 7 day pass for the 24th - 30th. Within Tokyo, the 2-3 day subway pass is a good money saver.
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03

thank you everyone!