Northern Kyushu itinerary for week of Dec 22?

We're heading to Japan again in December..and after a couple of weeks in Tokyo, we plan on spending the week of Dec 22nd in Kyushu. Our tentative itinerary:

Sun, Dec 22: Haneda to Nagasaki (via plane); Nagasaki overnight.

Mon, Dec 23 - Nagasaki overnight

Tues, Dec 24 - Nagasaki to Kumamoto (via ferry); Kumamoto overnight

Wed, Dec 25 - rent car in Kumamoto and drive to Mt Aso Park; Kurokawa overnight

Thurs, Dec 26 - drive to Takachiho; Kurokawa overnight

Fri, Dec 27 - drive to Beppu or Yufuin for overnight stay.

Sat, Dec 28 - return car in Oita; Oita to Haneda (via plane for one more week in Tokyo)

A couple of big questions:

from what i've read, December in Kyushu shouldn't be a huge issue in terms of winter driving...maybe a little more cautious around the Mt Aso area. Is this the case?

This being the Christmas week, will rental car locations be open?
Will the ferry b/w Nagasaki and Kumamoto be running? What about Mt Aso park and Takachiho gorge?

We are skipping Fukuoka, as my research tells me it's not absolutely necessary to visit - especially if you have limited time - and that the Nagasaki area is much more interesting.

What other area do you recommend to visit in place of Beppu or Yufuin? By staying the Kurokawa, i feel like we'll already have a really nice onsen stay. Anywhere else to add some variety to our adventures? And is somewhat close to a hub where we can fly back to Haneda the next day.

Xmas isnt a public holiday in Japan. Just a normal office day.

Hi. I missed Kumamoto unfortunately as we went very soon after the quake, so I cant speak for it. Of the places that I visited, I really liked Yufuin and would go back even if it was out of my way and quite liked Karatsu and would go back if it was on my way. I was not so keen on Nagasaki but possibly my fear of heights influenced that (it is almost more cliff than hill). If I hadnt been like a cat on a hot tin roof I probably could have gotten into it more. I liked Kurokawa and in particular Hozentai ryokan there but there isnt a lot to do in town.