Shinjuku - Kawaguchiko - 5th Station to Hakone - back to Shinjuku

Hi, I will be going to Japan next week and part of my itinerary is Mt. Fuji.

Is it only right to leave Shinjuku by 6am?
I know how to go to Kawaguchiko and 5th via JR Pass + Non JR Bus Line. But I can't find how to go to Hakone after 5th station.

My Itinerary for 1 day will be
6am - leave Shinjuku
7:30 - arrive in Kawaguchiko
9:00 - arrive in 5th station
11:00 - go to Hakone have lunch and go around

For Kawaguchiko, 5th station and Hakone - do I have to buy passes in advance? or is this places free for the public?

Please help.