1st time visit to Kyushu

Hi guys! My family will be visiting Kyushu for the 1st time during this January and we'll be flying into and out of Fukuoka. After some research, we have come up with a sample itinerary. Hope our questions can be answered along the way!

Day 1- Arrive in Fukuoka. Head to Nokonoshima Island

Day 2- Train from Fukuoka to Nagasaki. Visit bomb museum, peace park, and catch Mt Inasa view before catching night train back to Fukuoka. (The reason why we chose this instead of staying in Naga is because it would save us costs and there'll also be less hassle dragging out luggages to Kumamoto via Shimabara and Unzen.)

Day 3- Shinkansen from Fukuoka to Kumamoto. Visit Kumamoto Castle. (What are other things we can do?)

Day 4- Shinkansen from Kumamoto to Kagoshima and back. Visit Sakurajima. (Must we join tours for this trip?)

Day 5- Rent a car and drive to explore Mt. Aso, Takachiho, Daikanbo and Kurokawa Onsen. (Which area should we visit in order?)

Day 6- Drive to Yufuin and drop our car. (Are there any car agencies that allow dropoff at Yufuin and rental at Kumamoto?)

Day 7- Train from Yufuin to Fukuoka.

Which JR pass should we get then? Should we get the 3 day northern Kyushu pass and pay for the regular ticket to Kagoshima as the full Kyushu pass is double the price. We certainly welcome suggestions to improve our itinerary as well!

Day 2
Note that there are ferries from Nagasaki to Kumamoto

Day 3
There is a garden in Kumamoto. But mostly Kumamoto is a nice city to just walk about. You can see Kumamoto mascot character in one of the shopping malls at set times. Check tourist info for more info.

Day 4
Talk about a rush trip!
You won't need tours. you can take the island perimeter bus.

Day 5
Southern Aso is more interesting than North. If horses and quirky towns are your thing.

Yokagura is over rated, so you don't have to stay so late for that. In fact, Takachiho in my opinion is missable.

But if you want Takachiho, I recommend spending the night in Kurokawa.

Day 6
Unsure about Yufuin, but I am sure you can drop your car at Beppu.

There is a Fukuoka-Kagoshima return train pass. The rest of the trip, just use normal tickets.
Aso-Takachiho-Kurokawa

It sounds ambitious but Kyushu has something that will satisfy almost everyone.
For Fukuoka, Nokonoshima is a nice half day outing and the Island Park, while it has a stiff admission price, is one of the most beautiful places to explore with stunning gardens and views.
https://www.youtube.com/watch?v=H-RmFcmhs9c
There is far more to see and do of course. Ohori Park and Maizuru Park next door with the castle ruins makes a nice stroll.
https://www.youtube.com/watch?v=vbdQUjljOyw
https://www.youtube.com/watch?v=968CH00PboY
Nanzoin is also a spectacular temple with a great Buddha about 20 minutes by train from Hakata Stn and a must-see place:
https://www.youtube.com/watch?v=lT4VQFmHFyM
For a lot of other good info on what to see and do, have a look at:
http://www.thejapanfaq.com/fukpic.html
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...
Look into a Fukuoka Tourist City Pass also:
http://yokanavi.com/access_tourist_eg
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind.
http://www.ngs-kenkanren.com/eng/
http://www.at-nagasaki.jp/foreign/english/
http://www.youtube.com/watch?v=QdWRyHGW3rk
Be sure to see Mt. Inasa at night if the weather is clear, and try the Nagasaki chanpon and saraudon! Also the Koshibyo temple is an often overlooked shrine that is quite unique in Japan:
https://www.youtube.com/watch?v=Mu_JnkVInTU
Kumamoto is famous for its castle, Suizenji Garden (a must-see), Mt. Aso and Kurokawa hot springs.
http://www.visitkumamoto.com/
http://www.pref.kumamoto.jp/english/pdf/kankou_e.pdf
https://www.youtube.com/watch?v=K9xyXmU8U1g
That said ASAIK the Aso crater is still closed.
http://www.kyusanko.co.jp/aso/lang_en/
As an alternative, there's a beautiful botanical garden, Kujuhana Park just NE of Aso as well as Nabegadaki, a nice waterfall you can walk behind - it's NW of Oguni Town.
Kagoshima is great to see the live Sakurajima volcano, hiking, and hot sand baths such as on Ibusuki. For the volcano itself you can see a lot just exploring on foot, but there is also a one-hour fast paced bus tour that takes you to several places.
Plus the Senganen garden has a superb backdrop view of the bay and the volcano. See:
http://www.kagoshima-kankou.com/for/
https://www.youtube.com/watch?v=LxDmLuq6ssI
https://www.youtube.com/watch?v=lpnDK_Wux9Q
For comprehensive info on Kyushu, see:
http://www.visitkyushu.org