Day 6 - for a single day trip (of the two choices) I'd say Nikko.
Day 8 - I suggest arriving at Kyoto as late as possible (as long as you're not booked into a ryokan with evening meal included) so that you can spend more time in Takayama. It's a huge detour you're taking to get to Takayama, you may as well spend some decent time there.
Day 13 - I'd take out Naoshima. Miyajima is a large island with lots of undeveloped parts. Stay at a Ryokan on Miyajima and take it easy for a day - or use the cable cars to explore the upper parts, or just hike.
I propose - Day 12, arrive at Hiroshima, stay night on Miyajima. Day 13, explore Miyajima a little more and stay night on Miyajima or at Kurashiki (near Okayama). Day 14, vist Himeji castle before returning.
As for anything else.. I have reservations about suggesting MORE destinations for your itinerary, but if you are planning on going to Takayama from Tokyo, the town of Matsumoto is a lovely place to stop off at. Give it a minimum of 3 hours to get a brief glimpse of its main attractions, including its wonderful black castle (like entering a landlocked wooden battle ship).
And, of course, yokoso flyertalk, moniw!!!