Hi there,
It is really nice that you can come to Japan for such a long Holiday! Also the places you plan to visit include some of my personal favorites.
The trip from Matsumoto to Takayama is not very hard but it takes about 5 hours by limited express trainand involves just one transfer @ Nagoya station. If you don't want to be that long in the train you could perhaps plan a stay @ Nagoya, alternatively Kyoto is a good option too, as it only adds a minimum of extra travel time.
A visit to mount Koya?
I think this place is best visited by (married) couples to spend the night in a temple and enjoy the beautiful sights it has to offer. Of course if you just want to see the famous sights and enjoy the hiking trains you will be fine. To me it always felt like Koya was best shared with someone. Instead I would recommend a night's stay at Kinosaki onsen that can easily be reached from Kyoto by ltd. express train and you can enjoy some of Japans finest Onsen Culture.
14 Day Pass or 21 Day Pass?
I think a 21 Day Pass could be worth it for you. This for two reasons, NR 1 being that 14 days may just be a little too short for all your plans. If you used it on your first day of Shinkansen Travel (13th) it would be valid until the 26th, one day short until you go to Tokyo.
Reason number two would be savings, the difference between the 14-day JR Pass and 21-Day Pass is 12600 yen, basically if you can do this amount of travel in 7-days, it becomes worth the upgrade. With your side trips alone you would be over this amount compared to normal JR-Tickets and you get to ride the Shinkansen partly between Nikko and Kawagoe! Lastly the extra freedom you get to take any train you want for 21-days is well worth it!
Hope this helps!
-Daniel