Yakushima or Beppu, Yufuin and Nokonoshima?

Hi there, everyone!

Me and my fiancé are currently planning our honey money trip to Japan. Actually, we already decided what places we want to visit, but a few days ago, we stumbled upon the beautiful island of Yakushima - and now we are considering to change our plans in order to visit this place.

At the moment, our itenary in Kyushu looks like this:
-5ht of June - Beppu
-6th of June - Fukuoka, Nokonoshima
-7th of June - Fukuoka, Inunakitouge (I guess many people don't know this location, but it's a supposedly haunted tunnel in the mountains around Fukuoka city. For me and my fiancé, exploring lost places is a hobby we both share - also, I already went there, when I spent half a year abroad in Japan and I had a really great experience at this place, so I absolutely want to show it to my fiancé.)
-8th of June - Kumamoto and Mount Aso - and in the evening Kurokawa onsen
-9th of June - Takachiho
-10th of June - Yufuin
-11t-16th of June - Okinawa

Like I said, we are considering to change our plans in order to have time for visiting Yakushima. What do you think about this? Are there any places in our itenary that we could skip for this? At the moment, we're considering this: Skipping one day in Beppu, one day in in Yufuin (since we're already going to Kurokawa onsen) and maybe another day on Nokonoshima or in Takachiho OR one day on Okinawa (since we'll still have enough days to spend there), in order to head straight to Yakushima (and spend 2-3 days there) via Kagoshima, after visiting Kumamoto and Mount Aso. Do you think this is a good plan or would you reccommend us to visit Beppu, Yufuin and Nokonoshima instead of Yakushima? And if not, what do you think is more worth to visit: Nokonoshima or Takachiho?

That's it. I hope you can help us with this question and I'm grateful for any opinion about this!

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.com
http://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 through
http://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.

Thank you very much for this detailed description, Ken!
We now decided to skip Beppu and Yufuin (as we're already visiting Kurokawa onsen) as well as one day on Okinawa.

So, now our itenary looks like this:

-5th of June: Fukuoka (Nokonoshima, Fukuoka Castle, Nanzoin Park)
-6th of June: Fukuoka (Inunakitouge)
-7th of June: Kumamoto and Mount Aso - as well as Kurokawa onsen in the evening.
-8th of June: day trip to Takachiho - and in the evening, we get back to Kumamoto, return the rental car and take a late Shinkasen to Kagoshima
-9th of June: Travelling to Yakushima by ferry in the morning, leaving us with the rest of the day to spend there
-10th of June: Yakushima
-11th of June: Yakushima - spending the morning and noon on the island and taking the ferry back to Kagoshima at 13:30 -so we will arrive at 17:40 and can take a Shinkasen back to Fukuoka where we want to spend the evening
-12-16th of June: Travelling to Okinawa by plane in the morning of the 12th of June and staying there for almost 5 days.

We're going to rent a car for our journey to Mount Aso and Takachiho, as well as on Yakushima and on Okinawa. Do you think we have enough time on Yakushima or shoul we maybe consider to drop Takachiho off our list, too?

I'd say you're putting in a decent amount of time for Yakushima. If you're driving it makes things a lot easier and you can do a loop around the island if you like. The hikes can be very time consuming though - yet that is the real experience of the island - like being in some primordial world.
And if you are driving, you might look at getting the expressway pass to save some money.
http://global.w-nexco.co.jp/en/kep/

For Fukuoka, many good places are near a subway station, so a subway day pass might serve you well. Or look at the Fukuoka Tourist City Pass.
https://yokanavi.com/en/tourist-city-pass/
You could add more for June 6th - seeing the bricked up tunnel won't take a full day. You might like to see a whole lot of places listed on http://bit.ly/2nXwn4f someday but honestly to me those places are 95% hype. It's your time though. On a different direction, you might want to go see the Hakozaki Shrine also in Fukuoka - it has some historical value since it was burned down in the Mongolian Invasion in 1274, but your timing is about right to see the beautiful hydrangea garden they have in a big garden in the back. You can see some other ideas at
http://bit.ly/1KAm8cU

I'd keep Takachiho, in spite of the time it takes to get there and the overpriced boats to row through the gorge - it is quite a place.

-5th of June: Fukuoka (Nokonoshima, Fukuoka Castle, Nanzoin Park)

That is a pretty full on day since they are all in different locations and you'll need to you taxi/bus, ferry, the subway and JR lines.