Hi & Congratulations.
Yakushima is indeed beautiful, but it takes a lot of time and commitment. Not to mention more money, but this is your honeymoon and not the time to be a tightwad. You could fly from Fukuoka Airport direct to Yakushima, saving the most time and energy. Unfortunately the flights are not eligible for JAL's foreign tourist discounted fares.
These 2 sites have a lot of info on visiting:http://www.yakumonkey.comhttp://www.yesyakushima.com
If you are going to Kurokawa, then you could skip Beppu & Yufuin for other places, as you mention.
One big detail you didn't mention is how you plan to go around. If you're going to see the sights around Aso, then a car is ideal. Aside from the volcano (which you still can't go up to the crater) there is the Daikanbo Lookout, Kuju Flower Park, Nabegadaki Waterfall, Kikuchi Gorge, and others. You also have to take a bus or car to Takachiho. Given the time involved, an overnight there is probably a good idea.
Nokonoshima is quite different - it is just a 10 minute ferry ride from one of Fukuoka's wharfs and there is some hiking, picturesque bay views, and the Island Flower Park is also nice.https://youtu.be/H-RmFcmhs9c
It only takes a few hours and it's really not in the same category as Takachiho, which would fill up a day or more with all the travel. Really depends on what you want - you know your own interests best. For Fukuoka, you might browse throughhttp://www.thejapanfaq.com/fukpic.html
for a lot of other ideas. I would suggest you definitely go see the Nanzoin Temple - one of Japan's finest, yet unknown to the outside world and tourist guidebook "experts".
For Okinawa, to save some money you could try a low cost carrier like Peach.http://www.flypeach.com/pc/en
or use something like ANA's Experience Japan Fare.http://ana.ms/20Xtf8t
Plus if you want to go island hopping, one cheap way is JAL's Okinawa Island Pass.http://www.churashima.net/jta/company/islandpass_en.html
Since it's your honeymoon, you stay at some love hotels - there's a new experience every night.