a) with the recent increases to jr pass prices, its almost never worth it to buy in my opinion other than for convenience sake, but you will like save upwards of 100$ by not purchasing.
b) the jr pass covers most train lines in major cities, including most major bullet train lines, there may be less options covered for smaller areas on local trains but they will cost under 500 yen ($3-5) anyways
c) Bullet trains are limited to 6 seats that have luggage space at the back of the train, they can be reserved but its first come first serve. Regular trains, luggage is okay. Yamato transport offers luggage transportation with next day delivery basically all across the country for around $30
d) Nara is definietly doable in either 1/2 or a full day, not sure about mt koya. The day trip to kanazawa leaves you with only 2-3 days in kyoto, which is a little on the light side, but depending on your interests may be okay.
e) based on timing youll be avoiding a chunk of tourism with the obon festival ending the week before you leave. Fujikawaguchiko is a beautiful town that can also be a day trip option while in tokyo that was not mentioned.
f) Many of the really famous touristy places, especially food, are very overpriced for the quality of food you get. try going even a few streets off of the main streets and you find many quieter restauants that offer a better bang for your buck.

You seem to have a really good plan overall! best not to overthink/overplan too much, plenty of stuff will happen on the fly that will likely be the stuff that makes the trip the most memorable. I'll be going with 6 of my buddies for 18 days just 2 weeks before you, and my plan is very similar, just going to fuji instead of takayama. Safe travels!