Once again, we have a first trip to Japan that strikes me as being a bit overly ambitious. A few random thoughts:

Since you will be ranging from Tokyo to Hiroshima, with a lot in between, consider whether you can route your air travel to arrive in Tokyo (NRT) and depart from Kansai (KIX) outside of Osaka. That would save you up to 3 hours on trains and a hundred or more dollars in train fares. As Richard points out, Takayama and Matsumoto are much more accessible from Nagoya or even Osaka than from Tokyo.

Rather than 3 days in Tokyo and 2 days in Kyoto, I'd go with 2 in Tokyo and 3 in Kyoto - or even use that extra day you mention to make it 3 and 3. Don't cut Kyoto short.

Hiroshima/Miyajima is certainly worth more than one day combined. Consider a day in and around Hiroshima and another half-day on and around Miyajima.

Since you are going right past, I concur with Lap Lap's suggestion to see Kurashiki, but I would recommend Okayama for a stay. It is only 10 minutes by local train from Kurashiki but has a better hotel selection and some quite nice things to see, including a castle and one of the finest Japanese gardens. Kurashiki took me about a half day before I was ready to leave. I happily spent more than a full day wandering around Okayama.

Rather than shotgun so much into your one visit, I think I would take a more leisurely, in-depth look at the country. Perhaps arrive NRT and 4 days Tokyo/Nikko, 1 night in Nagoya to set up a daylong trip to Takayama and then on to Kyoto, 4 days Kyoto/Nara, 2 days Hiroshima/Miyajima, 1-1/2 days Okayama/Kurashiki, 1/2 day Himeji on the way back east, 1 day Osaka and depart via KIX.

My best advice is to avoid trying to do so much and go so many places that you miss the little things and lose the flavor of the country. Spend time wandering the "small-town-feeling" back streets and alleys. Walk into a ramen or soba shop and order whatever looks interesting. Roam the shopping arcades and temple grounds. Ride the subway, getting off someplace unexpected - I found the best senbei rice crackers I have ever had at a little sidewalk stand in Nezu when I was half lost finding my way back to the station. Don't spend all of your time on trains and checking off tourist sites.

JR