Juggling JR Pass and 13 Night Itinerary

Hi,

Please help! It seems like the more research we do, the more confused we get especially regarding the transport system!

Right, from the beginning...We have 13 nights in Japan arriving in the evening of August 12th.

On first planning, we thought:
4 nights - Tokyo,
1 night - Hakone/ Mt. Fuji
3 nights - The Alps (particularly to see the hot spring monkeys and do the Alpine Route), however unsure of the best location to stay!
5 nights - Kyoto (with day trips to maybe Nara/ Osaka/ Hiroshima).

Our questions are:
* Is this a do-able itinerary in this order? Or can you suggest a more time/ cost efficient itinerary covering these locations? * Can we travel from Hakone - The Alps - Kyoto or would it be better to travel from Tokyo - The Alps - Tokyo - Kyoto? * Is 3 nights ok to do the Alpine Route/ see monkeys? Can it be done in 2 nights? What location(s) would you recommend staying in? Can the Alpine Route be done in 1 day (for example a round trip instead of one way)? * How can our luggage be accommodated on transport...no backpacks but large and carry on suitcases! * would it be beneficial to purchase a JR Pass, if so which one and for 7 or 14 days? If 7, when would we need to activate it?

Any advice or guidance would be very much appreciated :-)

Thank you,
Rebecca