Green Car vs Regular JR Pass

Hi,
First, please let me say that I have never met anyone who has said that regular bullet train seats were uncomfortable or unsatisfactory, even for long trips. You can see the difference between the regular seats and service and the Green Seats here. Ultimately if the seats are worth the extra cost is a question only you can answer for yourself though.
As for your planned itinerary, there are Green Seats available on the Tokaido, Sanyo, and Kyushu lines - so I am not sure where you are hearing this from. There might be some rumour floating around that in Kyushu there are none, but you can clearly see that there are here. What is quite noteworthy is that your pace looks very hectic and you are not going to be able to see a lot in Tokyo and Kyoto. (Nara is also worth at least half a day). Hiroshima/Miyajima in a day is mostly possible if you start very early in the morning; some places open quite early and you can see some good places (Peace Park/Museum, castle, Shukkeien Garden) in the morning and spend the rest of the day for the real treat which is Miyajima.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. You can see this site for a good 1 day itinerary.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free.
You can also see a lot of previews on the best there is to see here.

For Nagasaki, there is no bullet train there (perhaps after another 10 years), so it will take 2 hours from Hakata Station in Fukuoka. In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, but if you don't have the time, some of the best sights are the Peace Park/Museum - but if you see the Museum at Hiroshima you might want to skip it - Glover Garden, Dejima, and the Koshibyo Shrine. Gunkanjima would also be well worth seeing, but it takes 3 hours and your schedule really doesn't allow it. But at night be sure to go see Mt. Inasa at night for a spectacular night view over the city.

Good luck.

Hello there,

Toraneko already gave you a very detailed run down of the details. What I can add to it from my own experience is that Green Class is the way to go if you wish to travel in extra comfort and quiet. Especially quiet that is, as generally less people travel Green Class and those that do also with the relax.

Normal class can somewhat be compared with Economy from airlines, although it is a bit better than most economy airlines. Theres a bit more space for instance.

As a general rule, going Green is nice if you can spend the money otherwise you'll be more than fine in normal class.

Hope this helps,
Daniel