If I were you, I would use highway buses all way around.
1. Haneda airport to Hakone (Gora)
Please see the timetable below.
2. Tokyo to Takayama
Take a highway bus from Shinjuku station to Takayama.
You can reserve a seat from the link below.
3. Kanazawa to Tokyo.
I would take a Shinkansen from Kanazawa to Tokyo.
If you are staying around Takayama area for a few days, I would suggest that you use Three-Star Tickets which includes a one way ticket from Shinjuku (Tokyo) to Takayama, a bus ticket to Shirakawa-go, and to Kanazawa.
This ticket is valid for 7 days.
You ca reserve the ticket at the following link.
I hope this information helps.
Have a nice trip in Japan!