Should 4 years old child pay the JR?

We are family of 3, 2 adults and one 4 years old child, we are planning to go to japan from 18-29September 2018 with general itinerary as follows:
19 sept from narita straight to osaka, strolling on minami (shinsaibasi, dotonburi, amerikamura etc), - our hotel will be at shinsaibasi osaka.
20 sept osaka-universal Studio Japan
21-22 osaka-kyoto + nara (visiting most temples and nara park) - back and forth trip
23 osaka to tokyo (strolling around tokyo like to shibuya, shinjuku, harajuku)
24 tokyo (roppongi, asakusa, akhirabara)
25 tokyo-mt fuji +hakone-tokyo
26 tokyo-disneyland
27tokyo-disneysea
28 tokyo
29tokyo-narita international airport