May 15-22 Japan Trip

Hey there - I just want to know the best option for our Japan trip this May. Our itinerary as follows:

May 15 - Arrive Osaka via KIX Airport
May 16 - Explore Kyoto
May 17 - Whole Day Nara
May 18 - Go around Osaka, check out and make arrangements to go to Tokyo (still deciding how to go to Tokyo from Osaka)
May 19 - Arrive Tokyo
May 20 - Whole day Mt. Fuji tour
May 21 - Explore Tokyo
May 22 - Head back to Osaka for evening flight to Manila

if we buy the JR pass (2 adults with 2 children ages 18 yrs old and 10 yrs old) does that include Shinkansen ride from Osaka to Tokyo, then from Tokyo to Osaka? Can we also use this pass to go to Mt. Fuji?

How much would be estimate of the JR tickets for all of us?

Thanks and looking forward to your reply.