Does 14 day JR pass make sense?

Hi Daniel-san,
Need your help, we are planning a 3 week trip to Japan starting Sep 26 as follows;

26 Sep to 3 Oct - Tokyo (staying in Shinjuku)
3 Oct to 5 Oct - Traveling to Kawaguchiko (understand we will can take JR from Shinjuku to Otsuki and then change to a private train)
5 Oct - 12 Oct - Kyoto but with day trips to Nara and Nagoya
12 Oct - 14 Oct - Leave Kyot, go to Himeji then onwards to Hiroshima
14 Oct - 19 Oct - Osaka with a day at Kobe and another at Universal Studios Japan
19 Oct - Fly out of Osaka Kansai to Singapore
My questions are;
1. Does the JR pass make sense for this kind of itinerary? if yes, would it be the 14 day pass? 2. How do we use the JR pass from Kawaguchiko to Kyoto (without having to backtrack up to Tokyo or Yokohama?
3. Would the JR pass cover the Osaka to Kansai airport and Narita to Shinjuku>
4. We have a 3 1/2 yrs daughter with us, does she travel free or we have to purchase a JR pass for her as well?

Thanks for your help.