Travelling from Tokyo to Fukuoka to Hiroshima and back

Hello -

We are arriving in Japan on June 2 and will stay in Tokyo for three days.

We plan to go to Fukuoka on the 5th.
From Fukuoka, we move to Hiroshima on the 9th
From Hiroshima, we move back to Tokyo on the 11th.

Thus, the total travel time would be June5,6,7,8,9,10, and 11.

My questions are:
1. Can we activate the JR pass on the 5th of June even if we arrive on the 2nd of June?
2. Based on my count, a seven-day (7-day) JR pass is sufficient; please confirm if my understanding is correct.
3. Please tell us which trains to take for the following routes?

Tokyo (Akasaka) - Fukuoka (Kokura Station) (June 5)
Fukuoka (Kokura Station) - Hiroshima Station (June 9)
Hiroshima Station - Tokyo (Akasaka) (June 11)

Many thanks and I will forward to your reply.