Actually ...

Are you planning to stay in Kyoto and do Hiroshima/Miyajima as a day trip? Same with Nara?

If staying in Kyoto the whole time, I don't see a problem with this itin. If staying a night or so in Hiroshima (recommended,) I'd suggest just running all the way down there from Tokyo, seeing what you want there and THEN heading to Kyoto/Nara. I really recommend 3 days for these - 2 for Kyoto and 1 for Nara.

I think I'd go at this a bit differently. I'd do Tokyo and Hakone up front and activate the Rail Pass for the last 7 days of the trip. Use it to run down to Hiroshima for a couple of days, then back to Kyoto for the rest of your stay. The 7th day, I'd use the Pass to run all the way back to Tokyo and on to HND for departure. No point in doing 2 separate stays in Tokyo. Of course, if your departing flight is TOO early, you'd need one last night near the airport but no point in getting there too early the night before. YMMV.

Trains run 7 days/week. Most tourist sites like Nara are fine on Sunday. The only problem there is that it is a day off for Japanese and sites can get a little crowded.