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Hi,
Either of your plans would work. I wouldn't get a 21 and a 7 day pass though. For day trips near Tokyo, you can get a 3 day Tokyo Wide Pass. Basing yourself in Takayama for 7 days makes little sense either - it is very isolated and it's worth a night or two to see the city and Shirakawago. But as a base, other cities would be better.

Kyoto to Koyasan would be a very long trip - it's worth staying overnight in Koyasan, and getting a Koyasan World Heritage Ticket. In the Kansai area you should also take a day and see Himeji with Japan's best castle. And if you like it, Hikone is not too far either. You can see Miyajima in a full day. Aside from the Itsukushima Shrine with the giant "floating" torii, the Daishoin Temple is also a must-see place. There are several other pagodas and temples, and climbing Mt. Misen offers a fantastic view over the island and whole area. If you have time, just a bit past Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle.

For Kyushu, you can actually get to Kagoshima from Fukuoka City in less time than to Nagasaki since it's a quick bullet train ride.
Kagoshima City has a number of great places to see, including Sakurajima and Senganen Garden, plus Shiroyama Park. The ferry ride going over was also particularly relaxing.
You can see both Beppu and Nagasaki in a full day - although getting from one city to the otherwould take you at least 4 hours. You might want to see the sights of Fukuoka as well - there are many good places especially the Nanzoin Temple Great Buddha which is a hidden gem that few have discovered.

Good luck.