Not sure you need 7 days in Kyoto itself, you may want to spend a few of those seeing Nara Park and other nearby places too. Nara has some unique sights, and you'll be close by.
Hiking in the Japan Alps - I did that too, though I was traveling the other direction (left Takayama, went hiking, arrived in Matsumoto that evening.) It is certainly possible to do in 1 day, and still get several hours of hiking in. Just be sure to check the bus schedule and be back from your hike in time for the last bus! Here's some shots of my hiking there:
Be sure to visit Matsumoto Castle in Matsumoto, and the Hida Folk Village in Takayama! Both are very interesting places!
Glad you're going to Koya-san, I think you'll enjoy it!
Kanazawa has an incredible park, very different from Nara Park. (Nara's isn't very ornate, it's mostly to house the herds of deer and provide a nice setting for the historical attractions and temples, while Kanazawa's is very ornate and beautiful) It is a great contrast to the wild beauty of the Japan Alps... every little bit of Kanazawa Park is perfectly maintained, like a life-sized bonsai scene. The top 2 photos below are from Kanazawa Park, so you can see what I mean:
Since you'll only need 1 day to get from Matsumoto to Takayama, you can stop and see Miyajima as you said you wanted. It's easily doable in 1 day.
You might also want to take a half-day to see Himeji Castle on your way there, it's widely considered one of, if not the best, castles in Japan. (Though if you go to Matsumoto Castle a few days earlier, you can decide whether you want to see another, even bigger one)
Be sure to eat the famous local Hakata ramen when you're in Fukuoka!
Overall, sounds like a great trip, I think you'll have a blast!