It doesn't look too bad at all. Five nights for Kyoto is pretty good - you could also go see Osaka in the evening (Dotonbori, Abeno Harukas Bldg, Shinsekai, Umeda Sky Bldg). Just make sure you add enough variety for Kyoto, since an endless series of temples and shrines will just make your eyes glaze over at some point. There are enough places to add to keep things fresh, like Nishiki Market, the Heian Shrine Garden, Murin-An, Shoseien, the railways museum, Iwatayama Monkey Park, Sagano bamboo grove, etc.
https://goo.gl/knQtMg For Hiroshima, if staying 2 nights you could add more. Take a day for the city, and a day for Miyajima. For the rest, you could see Iwakuni, or go see the nice Sandankyo Gorge if you like nature (75 min bus ride from Hiroshima Stn). For the city though, there is much more than just the Peace Park, and sadly many people cheat themselves by just leaving the city after seeing just there. The Shukkeien Garden is very nice, but there also is the secret jewel - the Mitakidera Temple which almost nobody ever finds. Plus try some Hiroshima style okonomiyaki at Okonomi-mura. Hiroshima Castle is OK but if you're seeing Himeji Castle you can skip it.
https://goo.gl/fAz4axNagasaki is a very interesting and lovely place to visit. Huis Ten Bosch may surprise you - it looks a lot like Holland (though I've heard some Dutch say it's much cleaner). Three nights is a good chunk of time though, so you could maybe include Sasebo, or see Unzen (the latter really almost requires a car though). Seeing Gunkanjima takes a lot of time (with just an hour on the island), and if the weather is bad, the boat is cancelled, so have a back-up plan just in case.
There is no rail access to Kurokawa or Takachiho, so renting a car is a good idea, not to mention to get to a lot of good places around Aso (Nabegataki Falls, Daikanbo Lookout, Yamabuki springhead, Kuju Flower Park, Kamishikimi Kumanoimasu Shrine, etc). Three nights at Kurokawa may be overkill though. Are you switching inns? Otherwise there is little you'll get that's new from the 2nd night.
For Kagoshima, Sakurajima can eat up a full day, but don't skip the really nice Sengan-en Garden as well as Amami no Sato. Simply breathtaking. Try the shochu liquor there too.
For Yakushima, logistics can be tricky, so do a lot of research. A full hike can take over 8 hours for some places. Renting a car can be a lot easier also. If you're going to Beppu, the Hells Tour only takes a few hours. You could also go to Oita City close by and visit the Takasakiyama Monkey Park, which is one of the better ones in Japan. You can also take the ropeway up to Mt Tsurumi.
Fukuoka is more famous for its food like ramen and mentaiko (spiced cod roe), but there is much more. Just outside the city is the Nanzoin Temple, which is excellent. Inside the city is also Tochoji with the nation's biggest wooden carved Buddha, plus the views from the Fukuoka Tower are nice, as well as the new TeamLab Forest exhibit by the Paypay Dome.
If you see Ishigaki, be sure to go visit the limestone cave there - it's like suddenly walking into another planet.
https://goo.gl/Q843Bg
