Driving to Takachiho, Aso, Kurokawa & Yufuin

hello guys,

Major help required~ Will be travelling to the Kyushu region in Dec and whilst constructing a route to the places of interest, i hit a snag. Can you advise if this route is feasible to do?

Day 1-4: Fukuoka (train) - Kumamoto
Day 5: Kumamoto (drive) - Mt Aso - Kurokawa Onsen (overnight)
Day 6: Kurokawa (drive)- Takachiho Gorge - Usuki Stone Buddha - Yufuin (overnight)
Day 7: Yufuin - Beppu - Hakata

Questions:
1. Renting a car from kumamoto for 3 days and returning it at Hakata will set me back about 35k yen, is this economically sound for 2 pax as compared to public transport? If public transport is more feasible, any advise on the route i should be taking in order to hit the above points of interest?
2. Is the drive in dec for those mountainous regions safe? This is our first time driving in the winter, not sure if there's anything to look out for.
3. Is Day 6 too rush? Looking to set off early in the day say 8.30am but i'm not sure how much time will be spent in Takachiho.

Thanks in advance!

For your first question, one of the things you should know is that there can be some stiff drop-off charges for putting the rental car at a different place than where you got it. But for the places you want to go see, a car can be very convenient and while 35K yen is pretty high, you also have the freedom to see a lot of places that you just can't see by bus at all if you choose.
For your first 4 days you don't give much info, but Hakata has some excellent places to see, especially the Nanzoin Temple.
http://www.thejapanfaq.com/fukpic.html
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...

I hope you are also aware that the area near the Aso crater is still closed off and you can't get near it. But if driving there are some great places you can see in the area - Daikanbo when the weather is good gives a fantastic look out over the whole area. The Kuju Flower Park is also excellent, plus the Rapyuta no Michi for another view, Nabegadaki for a beautiful waterfall, the Kikuchi Gorge for a nice forest stream walk, and more. If you want to see what you'd miss at the Aso Crater, see:
https://www.youtube.com/watch?v=Tyh_nKrv5tY

For Day 6 it is doable, although you won't be able to linger very long. It also doesn't take that much time to see the Usuki Stone Buddhas; but you will be on the road a lot. You might be able to fit in the Furendo Limestone Cave also - it is not Japan's best but it is pretty good. For Beppu there are of course its hot springs, but don't miss the Hells Tour and nearby monkey park/aquarium as well.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-hOggZamc2lUNTr46ApKX...
Driving in December is usually not a problem. In fact the first part of the month can still be fairly warm in the daytime, and you could see some autumn colours. The Kiyomizudera Temple in Miyama (not the one in Kyoto) was very good for the orange and gold leaves.
https://www.youtube.com/watch?v=jKAMjJ4VrTo
One big omission you have in your plans though is skipping Nagasaki. It really is the most interesting city on the whole island, and certainly worth it if you can fit it in.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-gbw2yXNLK92siHyNqSNx...
The food is also excellent. Real chanpon and saraudon are wonderful.