Help on how to get to these places in Japan

I'm proposing to my girlfriend on December 2017 and part of the surprise is a trip to Japan from December 10th until December 28th.

I have never been to Japan myself so I have been researching a lot on where should we go during December. We will arrive to Tokyo from Mexico and will get an AirBnB place for the whole timeframe so we can go to these places and come back to Tokyo for resting from time to time:

2 nights in Kyoto (trying to catch final days of Arashiyama Hanatouro)
The Oku-Aizu Tadami Line trip (and spending a night there, then going back)
Spending a night at Minakami Onsen (specifically Takaragawa Onsen)
One day visit at Mt. Fuji.

I think buying the JR pass is a wise decision but I'm not sure if I will be able to reach all of these places just with that or I will have to take into account some other train/bus tickets. Would anyone be willing to help me privately in figuring out how to get to each AirBnb location where we will be staying?

I usually use Google Maps to find out the travel options and prices, looks like everything you picked is accessible via JR:

Tokyo - Kyoto by shinkansen
The Oku-Aizu Tadami Line is operated by JR and accessible via either the Tohoku or the Joetsu shinkansen (http://japan-magazine.jnto.go.jp/en/1510_okuaizu.html)
Minakami Onsen is accessible by a combination of shinkansen and local JR line.
Note that the shinkansen are also operated by JR.

As for a visit to Fuji-san in December, it is probably best to pick a location with a famous view of Fuji-san.

Whether or not this is enough to warrant a 2 week JR pass I'm not sure, a 1 week pass should be fine with your trip to Kyoto. Use Google maps to find out the prices and compare them to the cost of the JR pass. If they're close you're better of with the pass as you'll find yourself also using the pass to get around Tokyo and those daily trips do add up.

Thanks a lot! That was very useful.