Hi Ephrat,
Regarding 1st / 2nd November holidays, there should not be an issue with major bookings in trains or hotels. Maybe some hotels particular to a local event are full, but trains no problem, according to my friends. E.g., book hotels first, then the trains!
If you want to maximize the first class tickets, choose limited express trains if they are available. On HyperDia, you can tell if it has first class by clicking on the pull- down tab under "Seat Fee." If that particular train has first class, it will appear on the menu of choices as "Green seat." Otherwise, it will be limited to reserved and unreserved, and since you will pay nothing extra for a reserved seat using a JR pass, you should get a reserved seat when picking up the tickets from JR.
On the particulars of your planned journeys, yes, you will have to take some non- JR lines and just pay the man. (The JR pass is still a great deal, if you do the math - there is no pass covering all Japanese rail companies. If you ever find one, take a look around for the angels, because you have reached heaven.) Very limited choice for green seats to Koyosan, at most a limited express for part way and then change trains. Not worth the bother in this case.
Your route from Koyosan to Karatsu is pretty much as you describe. SHinkansen from Shin-Osaka to Hakata, green car all the way. You could take other routes and depending on several factors, shave off a few minutes or a few yen, but -- not worth the effort. From Hakata to Karatsu has some very nice scenic moments (and some boring ones as well.) Getting from Karatsu to Kagoshima - well, I played with that a bit. My first instinct was to go from Karatsu to Kubota, then Saga, and pick up the Shinkansen there, but with the JR green pass, going straight back to Hakata may be the ticket. You can play with all the times and connections and make your own choice (it's a fair bit of work at times, which I don't have the time for right now, sorry.) My quick look said that there was no good choice at all - but least of evils was but to retrace your route to Hakata and take the Sakura Shinkansen to Kagoshima.
Now that I know your route, I have one more recommendation: Make a stopover in Shimonoseki and go to the fish market. The fishmongers there are ever so more approachable than those at Tsukiji (who have tried at various time to ban all tourists.) The Deal of the Century at this market is Fugu. Yes, blowfish / puffer fish / in all its names. Do not be put off by the many reports of deaths by pufferfish. This is all media hype around a small number of cases every year where the fish were prepared by unlicensed chefs. If these same reporters wrote about your hometown, you would never go outside your door, day or night. Anyway, you can can a complete fugu "setto" (set meal) at lunchtime for 1,200 yen. Yes, $12.00, ridiculously cheap compared to Tokyo prices. You'll be there at the right time of year, too.
Kokura is also an interesting spot with great nightlife. A fun variant would be Shimonoseki > boat to Mojiko, some nice
restaurants including an excellent brewpub > local train to Kokura . Shinkansen on in to Hakata. (You can also walk UNDER the straits at Shimonoseki, 40m under the water via a pedestrian tunnel. Now we are getting way beyond what you asked for.)
Don't you wish you had three months for this one?
Have a great trip and if possible reply- to when it's all over and let us all know how it went :-)