Hi again and welcome back!
1.) The best way to visit Fuji is indeed via Tokyo and it can be hard to go directly from there to either Osaka or Kyoto, the places you are planning to visit there are all part of the Hakone Area.
What I recommend is getting a Hakone Free Pass that you can use to travel on almost every kind of transport in the area and it is very easy to make good savings with. Once you are done sightseeing in Hakone I would recommend traveling to the station called Odawara with the Hakone Free Pass and from there take the Bullet Train with the JR-Pass to Osaka.
2.) You can the JR-Pass to travel between Osaka and Kyoto and on local trains in Osaka with the JR-Pass. Other kinds of transportation like the metro in Osaka unfortunately are not covered. For Kyoto I would recommend an all day bus pass that is only 500Yen.
3.) You can reach Toyota Kaikan Museum from Nagoya in 1.5hours. Nagoya is on the route between Kyoto and Tokyo but the 1.5 hours would be extra travel time. Also refer to the access page from the museum.
Hope this helps,
Daniel