I am afraid your itinerary is utterly impossible. It contains multiple non-existing train rides and is generally way too packed.
1)
train overnight (moonlight nagara)Seat reservations are mandatory on the Moonlight Nagara. There is a high probability that the train will be completely booked out several days before travel date. Make sure to secure a seat reservation in advance, otherwise, there is a high probability that you won't be able to ride the train.
Hiroshima, train over to Fukuoka? (moonlight kyushu?)The Moonlight Kyushu does not exist anymore.
Hike Kirishima
Kagoshima
down to YakushimaToo packed for a single day. The last ship to Yakushima is at 15:50. The last flight from Kagoshima Airport (close to Kirishima) is at 17:00.
A rental car is recommended for Kirishima National Park, as public transportation is virtually non-existent for access to the hiking trails:
http://www.japan-guide.com/e/e4626.htmlNight train back to TokyoThere is no night train between Fukuoka and Tokyo! But there are overnight buses.
First of all, I would like to keep it as close to 10 days as possible, so I'm thinking of taking out either Kirishima or Mt Aso. Preferably Aso, but any suggestions on which would be better?As you can see, your itinerary contains some major transportation related problems. You have to solve them first of all. For time efficient travel, I recommend to fly (with an air pass). And to use a rental car in the regions where public transportation is bad or non-existent.
For example:
Tokyo - (air) - Beppu - (rental car) - Aso - (rental car) - Takachiho - (rental car) - Miyazaki - (rental car) - Udo Shrine - (rental car) - Kirishima - (rental car) - Kagoshima Airport - (air) - Yakushima - (air via Kagoshima Airport) - Tokyo
Would it be possible to see Miyazaki and the area all in one day, rather than 2?No, it is impossible. Miyazaki is one of Japan's largest prefectures and transportation is slow.
Also I was told that the Moonlight Nagara will be running beginning the 13th of March, is that the same for the Moonlight Kyushu?No, the Moonlight Kyushu has not been operating in years. Just wondering: where do you get the information about the Moonlight Kyushu?
Lastly, would it be crucial to book rooms in every city, or will I be alright just showing up on the day in some smaller towns?Just showing up without reservations translates into time loss during your trip and an uneasiness about not being sure whether you will find a room in your budget range. In the worst case, you will find yourself without a room at all. I recommend to make all reservations before you leave for Japan or at least one day in advance.
Also, can you stay the night at train stop waiting rooms or something, or will they kick you out?At some stations it might be possible, but the locals will not appreciate it. It is not an acceptable thing to do in Japan. So, I strongly discourage any foreign tourist to behave like homeless people in their host countries and plan their trips responsibly.