Fukuoka Itinerary

Can you please confirm if this is doable? Feel free to suggest other places for visiting

Day 1 (May 9)
- arrival in Fukuoka (8pm)
- go to hotel

Day 2 (May 10)
- Kushida Shrine
- Ohori Park
- Fukuoka Castle
- Fukuoka Tower
- Momochi Seaside Park

Day 3 (May 11)
- Dazaifu Tenmango shrine
- Ichiran ramen (lunch)
- Tenjin shopping center
- Canal City

Day 4 (May 12)
- Nanzoin Temple (Reclining Buddha)
- Uminonakamichi Seaside Park
- Yatai food stall

Day 5 (May 13)
- travel to Hiroshima
- Hiroshima Castle
- Atomic Bomb Dome
- Children's Peace Monument
- Cenotaph for A-bomb victims
- Peace Memorial Park
- Hiroshima Peace Memorial Museum
- Shukkei-en Garden
- overnight in Hiroshima

Day 7 (May 14)
- ferry to Miyajima
- Itsukushima floating torii gate
- Itsukushima Shrine
- Daisho-in Temple
- Momi-dani-koen Park
- Mt. Misen via Miyajima ropeway
- ferry to Hiroshima
- travel to Fukuoka

Day 8 (May 15)
- Kyushu tour
- Hells of Beppu
- Mt. Aso

Day 9 (May 16)
- free day

Day 10 (May 17)
- shopping
- flight back home

Day 8 is not gonna happen. Too much. All others days are good.

I'd say it looks pretty good. For May 10, you might want to put the Fukuoka Tower last. The best time if the weather is good is to go up just before sunset and watch the city change from day to night.
For May 11, you could add more if you like. For Dazaifu, be sure to see the shrine garden in addition to the shrine. You could if you want continue farther south to Yanagawa and ride the nice canal boats there. It's a bit unclear how much shopping you want to do, but the stores are open til 8 PM.
The Nanzoin Temple you have for May 12 is one of the highlights of Kyushu - yet still mostly unknown to most. It's also just a minute or two walk from the train station.
https://goo.gl/X3LDaE
http://www.thejapanfaq.com/fukpic.html
Getting to Uminonakamichi will be a bit time consuming though. You can get there by train, but the ferry from the Bayside Place pier might be more enjoyable.
For Hiroshima, a lot of the places you mention are all together. The Shukkeien Garden is very nice, but if you can fit it in, the Mitakidera Temple there is also an amazing sight and worth your attention.
https://goo.gl/fAz4ax

You also might consider seeing Iwakuni on the way back from MIyajima to Kyushu.
https://youtu.be/0COOZXi66mM

The only snag I see is trying to see both the Beppu Hells and Aso in one day.