You will get your money's worth and then some! Use the pass for the last 14 days of your trip and then to go to Narita Airport on the Narita Express. For getting to Tokyo, look into Keisei/Metro package or use a 1000 yen bus and 3 day subway pass.
If you want to add more value, consider going to Nikko for a day using your pass on one of your final days in Tokyo. It is well worth seeing.
For your Day 9, Arashiyama is part of Kyoto and one of the few places you can use your JR Pass in Kyoto to get to. Don't miss the temple at the north end, the Otagi Nenbutsuji, which is one of the most humorous and whimsical places in Japan you will ever visit. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails.
For Himeji, the castle is the best in Japan but don't miss the Kokoen Garden next to it, and if you have time, go to Mt. Shosha.
For either Days 11 or 12, if you have time, just a bit past Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle. One other gorgeous but overlooked place is Akiyoshido which is a 300 million year old limestone cave 1km deep with an underground river and giant caverns.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free.
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here. While it has many good places of interest, 3 days may be more than enough, and you could expand your plans a bit and see Nagasaki as a day trip if you start early, or see Kumamoto for its castle and wonderful Suizenji Garden. Beppu is another possibility and famous for its hot springs, but its Hells Tour, plus the Takasakiyama monkey park/aquarium are all terrific. The best hot springs though you'd need to take a bus to get to, but Kurokawa in Kumamoto is one of Japan's finest hot spring resort areas with beautiful forested outdoor baths.
Going all the way from Fukuoka to Tokyo in a day will be long - you could of course break up the trip with one of your planned stops in between.
Looks like a very rich and varied itinerary for you.
Best of luck.