Questions and Decisions to make: 7 or 14 days?

Like most people here, it seems, I'm trying to decide between a 7 and 14 day pass for our trip. Here is our itinerary:

Day 1: Narita Airport to Tokyo
Days 2-4: In Tokyo
Day 5: Travel to Hiroshima
Day 6: Travel to Osaka
Day 7: Travel to Kyoto
Days 8-9: In Kyoto
Day 10: Travel to/near Hyakawa
Day 11: Travel to Tokyo
Days 12-15: In Tokyo
Day 16: Travel to Narita Airport

To me, it does seem like 7-day passes will work for us. However, I don't know what the rates are to travel the trains inside of Tokyo. Are they expensive enough to warrant buying 14-day passes?

Second, how do I activate my pass for the day I want it? Do I choose that day when I purchase it, or do I activate it online/somehow on the day we need it during our trip in Japan?

Finally, what is the best way to get from Kyoto to Hayakawa? Should I take the shinkansen all the way back to Tokyo, and then backtrack, or is there a stop near Hayakawa?

Thanks so much for all the help!