Boy, where to start?
First, the idea to fly into KIX and out from NRT is good. It saves a huge backtrack.
If you wind up going from Osaka to Hiroshima/Miyajima to Sendai to Tokyo, then the JR Rail Pass is definitely worth it. Understand that you must purchase the Pass exchange vouchers before you go to Japan and exchange for the actual Passes after you arrive. When you make the exchange, you will specify what 7 consecutive days it should cover. That should include the longest train trips. There are local area Passes you can purchase after arrival that will help with local area travel in the Kansai region.
http://www.accessjapan.co.uk/railpass/regional.html
A Suica prepaid subway/train card will cover Tokyo.
I would suggest shortening your Tokyo stay by 1 day and adding a night in Sendai so you have a full day ( plus 2 nights) there. The best places to visit will require some train time to reach. I may be the only one here, but I found plenty to keep me busy there for 3 nights/2-1/2 days. From Sendai, it is a short local train ride to Hon-Shiogama, where you can catch the bay ferry to Matsushima. The ferry ride is nice and quite scenic and the little town of Matsushima has several beautiful places to see on a walking loop. From there, you can take the local train back to Sendai. Another day trip from Sendai is up into the mountains to Naruko Onsen, where, in addition to the hot springs spa industry, there is a nice river gorge hiking trail. I have no idea what it will be like in July but it was spectacular in fall.
If you go with the 7-day Rail Pass, activate it to be in effect for your trip to Hiroshima. From Hiroshima, it will also cover the JR train ride to the docks and the JR ferry to/from Miyajima. If you can see your way clear to coming up with another night in Hiroshima, I'd say see the A-Bomb museum and Peace Park on your arrival day, spend the night and then go to Miyajima the next morning, spending the day there. JMHO.
Make sure the 7 days of your Rail Passes cover the trip to Hiroshima, the long run (5.5 to 6 hours) from Hiroshima to Sendai and the run back down from Sendai to Tokyo. Anything else is gravy. For example, if you are only using the Pass for 5 or 6 days to cover the long trips, you could activate a day or two early and use the Pass for your trips from Osaka to Kyoto or Nara. You want to maximize the value of that 7 days.
I encourage you to simplify and figure each days' destinations will take more time than you think. You may have more activities planned than you can comfortably do in a relatively short time. June is the monsoon (rainy) season, although by the end of the month that should be clearing up. Think about what you will do with luggage while you are relocating and touring nearly every night. Sometimes it is better to plan a full day/2 nights in a new place so you can just leave everything in your room while seeing the city.
You can do all of what you have planned but will be traveling long periods on a few of your days. The shinkansen is comfortable but plan ahead with snacks, lunch and/or drinks on the longest days. They have a snack cart but the young women manning them rarely speak English and tend to blow through quickly. For long travel days, I like to shop for a nice obento box lunch at the station before leaving so I can browse the offerings at my own pace.
I cannot help you with dietary accommodations. Avoiding beef won't be much of a problem but much of the best Japanese cuisine is fish-based - or pork and chicken - and too good to pass up. Again, MHO.
For calculating train trips/times, Hyperdia is the tool of choice:
http://www.hyperdia.com/classic_en/hyperWeb.cgi . If using the JR Rail Pass, don't forget to check the box to "Exclude Nozomi." That fastest shinkansen is not available to Pass holders. The next fastest "Hikari" trains are just as fast, they just stop at a few more stations. I prefer to avoid the Kodama shinkansens as they stop at almost every little station along the way but they are also an option.