(1) Hakone is close enough to Tokyo that a JR Pass is a waste of money. I believe Odakyu offers reasonable Hakone-area passes since they’re a major train operator in the area, so you could look into that. Within the city, just buy a fare card (Suica or Pasmo) and fill it as needed. Subway hops are cheap.
(4) Hakone’s pretty spread out, but there are buses.