Agree with txflyer. Go to Tokyo as soon as you get to Jpn. If you're visiting NYC for 4 nights, you wouldn't spend the first 2 nights next to EWR even if the hotel were free, would you? In particular, staying put in NRT to do a day trip to/from Kawaguchiko or Kamakura makes no sense.

You could do Kawaguchiko or Kamakura as a day trip from Tokyo, but not both. You could probably do either Nara or Kobe as a half-day trip, as long you're not intent on trying to see everything. Each could easily justify a whole-day trip. Personally I love Kobe, although I'm sure many on here would choose Nara.

I'm a big advocate of minimizing the number of times you check in/out of hotels. If it were me, I'd do the following:
Nights 1~4: Tokyo hotel - spend 3 full days in Tokyo, or use one of those days to take a day trip to Hakone, Nikko, Kamakura or Kawaguchiko.
Day 5: Get to Kyoto mid-day.
Nights 5~7: Kyoto hotel - you have 3.5 days to play with. Spend 1.5~2.5 days in Kyoto; if 1.5 days in Kyoto, then you have time to go to Nara or Kobe. You then have 1 day remaining to go do a day trip of Hiroshima+Miyajima.

I do think you have time to do Hiroshima+Miyajima, and it's well-worth it. But, particularly with your tight schedule, I recommend it as a day-trip. It'll be a long day trip, but definitely doable without feeling rushed. I've done it a couple times from Kansai, no problem. Plan on spending ~3hrs in Hiroshima, 5~6hrs in Miyajima.

If you want to book with points, you may have better luck finding a hotel in Osaka rather than Kyoto. Those cities are close together, and they'll both serve as a good base for visiting Kyoto, Nara, Hiroshima, etc.