Jr pass cover

Hi,
Yes, JR will cover all your routes. But you may not be aware of the long rides in store for you. Hiroshima to Hamada is 3 hours with a good connection, and returning back to Osaka will be 4½ hours minimum. Tokyo to Sapporo will be between 7¾ to 8½ hours of exhausting riding. Since you are blessed with a very long time to see Japan, why not stop off somewhere along the way? There are many good places, like Kakunodate, Hirosaki, Hakodate, and so on.

For Mt Fuji, you need to choose between seeing Hakone or the Fuji 5 Lakes area - or see both. The JR Pass can get you part way there, but then you can save money on one of the passes by Odakyu.

While you're in the Kansai area, you should include some time for Nara also.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free!
You can see both Hiroshima and Miyajima with an early morning start. You can cover the Peace Park/Museum, castle, Shukkeien Garden and Okonomiyaki-mura, and then go down to take the JR ferry to the island. Aside from the Itsukushima Shrine with the giant "floating" torii, the Daishoin Temple is also a must-see place. There are several other pagodas and temples, and climbing Mt. Misen offers a fantastic view over the island and whole area.

If you have time, just a bit before Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle.
In Tokyo, you can get around cheaply on a 2 or 3 day subway pass, and to/from Narita there is the 1000 yen bus.

Good luck.