1

I would not bother with Nagoya on a short trip like yours.

The Hakone area and the Five Lake area are somewhat far apart, so checking in to a hotel at Hakone and then going to the 5th station will waste lots of time. Also, if you want to see both Hakone and five Lakes you'd need a minimum of 2 (mostly) full days in my opinion. Unless you are very much into Mt Fuji (who might not be visible in June anyway), on a short trip to Japan, I'd spend at most one day on it, by going to either Hakone or Five Lakes.

Also the 5th station is basically just a couple of souvenir shops on a mountain slope and you can't see the typical shape of Mt Fuji from there. IMO better to see the mountain from a distance, e.g. from Mt Tenjo or the lake shore if you go to Five Lakes, or from Moto Hakone or around Owakudani if you go to Hakone.

Transport to Mt Fuji is explained at http://www.japan-guide.com/e/e6901.html

With this itinerary a JR pass won't make sense. You might want to look into the Hakone Free pass for local transport in Hakone instead.

2

Do you have exact dates?


What time does your flight arrive?

It is relatively easy to get from Nagoya to Kyoto so I would probably go straight to Kyoto.
http://www.japan-guide.com/e/e3302.html?aFROM=2155_2158


I would probably just visit Hakone on the 4th day, then visit Fuji Five Lakes on the fifth day, then go from Fuji Five Lakes back to Tokyo. This will be much more efficient I think.

Which hotel/ryokan do you stay at in Hakone? You may be able to use the luggage delivery service from Hakone-Yumoto Station to your hotel (same day service). Alternatively, you may wish to ship your luggage using takuhaibin from Kyoto to your Tokyo hotel and just take an overnight bag with you to Hakone, this might be the best option:-
http://www.japan-guide.com/e/e2278.html


Hakone to Kawaguchiko:-
http://www.japan-guide.com/e/e6905.html?aFROM=5200_6900

Kawaguchiko Fifth Station:-
http://www.japan-guide.com/e/e6922.html


Yes.


http://www.japan-guide.com/e/e5209.html


Definitely not.