here is my final itinerary, abit relaxing.. alot of chilling time and do some shopping.
Day 1 - Fukuoka and transfer to Beppu (activate 3 days Northern Kyushu pass), visit Kitsuki
Day 2 - Beppu, day trip to Yufuin, Beppu ropeway
Day 3 - Beppu, Jikugo tour, try jigokumushi and maybe sand spa
Day 4 - Beppu to Kagoshima (activate 5 days all kyushu pass), maybe travel to ibuski
Day 5 - kagoshima, go Sakurajima
Day 6 - Kagoshima to Kumamoto
Day 7 - Kumamoto, castle visit
Day 8 - Kumamoto to Nagasaki, go Huis ten bosch
Day 9 - Nagasaki, Gankanjima
Day 10 - Nagasaki to Fukuoka, stay in Fukuoka until we leave on Day 13.
Decided not to do Mt Aso because the ropeway is not fully in operation.
Hi VD, I've just return from Kyushu trip. Here's my inputs regarding your itenaries.
Beppu jigoku: personally, I think not worth going. I've went for blood and tonardo and decided to skip the rest. would suggest going on a day trip to usuki Buddha statue instead.
Nagasaki, gankanjima: think you'll need to book gankanjima in advance. I've tried to book about 1-2 mths before my trip and wasn't able to get a slot as it was fully booked

You could only visit gankanjima ran by one of the 3 tour operators.
If you have time, would recommend going unzen jigoku which is within Nagasaki prefecture. There's an express bus that goes there from Nagasaki bus terminal but only have 3time slot. IMO, unzen jigoku is much better than the 2 beppu jigoku I've visited and moreover, admission is free
Fukuoka: i've splited shopping into 2days of about 4hrs each day. Have went tenjin on first day and hakata canal city on 2nd day. regretted that I did not allocate more time for shopping. 4hrs is not enuff >.<
Need at least one day on each location. Do note that shops closes mostly ard 8pm-9pm. Restaurants closes mostly ard 11pm-3am.
Kagoshima, sakurajima: would suggest you to make this itenary flexible. I have to change my itenary a bit as it was raining on the day I was suppose to visit sakurajima.