Would a Japan rail pass be useful for our trip

Hi Daniel-san

We were looking at getting a 7 day rail pass to use for part of our trip. Our itinerary is below:
Arrive in Narita on 27 October and travel to Tokyo from the airport.
29 Tokyo to Hakone
30 Hakone to Kyoto
01 Kyoto to Osaka
03 Osaka to Naoshima
04 Naoshima to Hiroshima
07 Hiroshima to Narita

Thinking it would be good to use the 7 day pass from 29th Oct-4th November or would it be better to get a 14 day pass?
Also can we activate the card whenever we like.

Thanks in advance,
Erin