Review drive plan in Kyushu

Hi All

As I will have a trip to Japan next year with my family, we have a plan to rent a car. Below is a plan,

1 HAK
2 HAK-HTB
3 HTB-NAGASAKI
4 NAGASAKI
5 NAGASAKI-KUMAMOTO
6 KUMAMOTO-TAKACHIHO-ASO
7 ASO-KUROKAWA ONSEN-BEPPU
8 BEPPU-Yufuin-Beppu
9 BEPPU-HAK-HIROSHIMA

Pls help us review if this plan is doable. and I have few question as
1. On day 6, do we have enough time to cover this route? If we leave Kumamoto around 10AM, when should we arrive ASO?
2.Is Kurokawa a worth visit? is there any interesting point other than onsen?
3. For day 7-9, should we also visit Oita? or we should go only Beppu and Yufuin?

Actually, we need to go Kagoshima / Miyazaki but we do not have enough time. Because we have plan to go Hiroshima for 3 days. Should we cut some day in Hiroshima and try to go Kagoshima?

Thank you so much

Pls help us review if this plan is doable. and I have few question as

Looks good to me. Except day 9 which involves too much driving. Is it really necessary to return to Hakata once? If not, than I recommend to take the ferry from Taketatsu to Tokuyama:
http://www.suonada.co.jp/c_searoute.html

On day 6, do we have enough time to cover this route? If we leave Kumamoto around 10AM, when should we arrive ASO?

It depends how much time you want to spend in Takachiho. As for the driving times, Google Maps has become quite good at estimating them these days:
https://maps.google.com/

2.Is Kurokawa a worth visit?

Yes, definitely. Nice town, great rotemburo. Overnight stay recommended for full enjoyment.

is there any interesting point other than onsen?

Not really.

3. For day 7-9, should we also visit Oita?

Depends on what you want to see in Oita. I personally do not know of many attractions in the city. What do you want to see there?

Actually, we need to go Kagoshima / Miyazaki but we do not have enough time. Because we have plan to go Hiroshima for 3 days. Should we cut some day in Hiroshima and try to go Kagoshima?

Kagoshima would add a lot of driving time to your itinerary, so I would not recommend it too strongly. On the other hand, Kagoshima is a great city with an amazing volcano... You will have to make the decision.