Kagoshima-Fukuoka in 4 days? How?

Can someone comment on the following plan?
What should I change? Any recommendations?

Day 0. Arrive in Kagoshima around 9pm. Stay somewhere simple (toyoko-inn?)

Day 1. Early morning - visit Sengan-en gardens, then take a ferry go Sakurajima, walk there for an hour, come back, then pick up a rental car, drive to Chiran, see kamikaze muzeum (are their English explanations?) and the samurai gardens (?), finish about 3-4pm, drive to Ebino Kogen (~2 hours), stay there overnight. (where exactly? can you recommend a place with a rotenburo?).

Day 2. After breakfast, walk around a lake, then drive to Aso (2 hours), walk around (any specific route recommended?), then drive to Kurokawa Onsen (1 hr), stay overnight (where exactly? can you recommend a place?)

Day 3. Drive to Hita (what is there? I don't care much for pottery) (1.5 hours), get on an expressway, drive to Nagasaki (3 hours) , spend evening in Nagasaki (go up the mountain?).

Day 4. Spend half of day in Nagasaki, then drive to Fukuoka (2hr), walk around Fukuoka for couple of hours, return car, get on a bus out of Kyushu at 9pm.

... or another option: after Aso, instead of going to Kurokawa, drive to Kumamoto, take ferry to Shimabara, stay overnight at Unzen, then, in the morning drive to Nagasaki. What do you think? Is Kurokawa much better than Unzen?

How long would it take to drive Aso-Kumamoto?

Day 1 is very busy. I recommend to get a rental car from the morning or - if the rental car outlet is still open - at Kagoshima Airport.

The rest of the trip is also very fast paced. You will be spending a lot of time behind the steering wheel. But if you don't mind that and prefer a fast pace of travel, it can be enjoyable.

Ebino Kogen (~2 hours), stay there overnight. (where exactly? can you recommend a place with a rotenburo?).

I am only aware of the following simple accommodation, which does have a rotemburo:
http://www.ebinokogenso.jp/

What do you think? Is Kurokawa much better than Unzen?

I haven't been to Unzen yet, but Kurokawa ranks as one of my top onsen destinations in all of Japan for its great baths and nice town.

How long would it take to drive Aso-Kumamoto?

About an hour.

I recommend to get a rental car from the morning or - if the rental car outlet is still open - at Kagoshima Airport.

I am not sure I understand why. I don't really need it at Sakurajima, so why take it there?

I know it is fast paced. However, the starting and ending points are fixed. If you can think of any other suggestions in between, I'll be greatful.

I am not sure I understand why. I don't really need it at Sakurajima, so why take it there?

Because unless you take a taxi, you will spend a lot of valuable time accessing Senganen and then traveling from Senganen to the ferry terminal. Have you looked into public transportation around Kagoshima? It is not efficient at all for access to Senganen and the ferry pier. And then from the ferry pier to the car rental outlet.

A rental car, on the other hand, will allow you to move very flexibly and quickly between each site and even allow you to drive around Sakurajima a little bit.

Perhaps you are correct. I was thinking about a taxi as they are pretty cheap in Japan.
Do you think it makes sense to take a car to Sarukajima (and pay for the ferry)? Is there anywhere to drive?

You will not see much without transport on Sakurajima. And who on earth told you that taxis were cheap in Japan.

I know about taxis from experience in Kyoto and Tokyo, they rarely cost more than $20 for distances where NY rate would be $40-50.

In description of Sakurajima it says that there is a trail close to ferry. Where would you suggest going on a car there?

In description of Sakurajima it says that there is a trail close to ferry.

It is nothing outstanding.

Where would you suggest going on a car there?

My top recommendation for Sakurajima is Furusato Onsen, but it requires more than just an hour. Just circling the peninsula is also mildly interesting, but probably also requires more than an hour. There are some look out points, but they seem to be out of business and not anything special.

Frankly, Sakurajima would be the place to skip if you want to slow down you pace on day 1. The mountain is nicest from afar in my opinion.