First, sorry if I keep pushing flying, it's not just because it saves time. I admit, the two times I have flown within Japan the flights have been absolutely gorgeous in terms of scenery. I just really enjoyed flying over Towadako and the Shimokita Peninsula a lot. Last trip I flew from Narita to Itami to get to Kyoto. I did actually save money and time doing it and the view of Fuji-san was the best we got that trip. I also just have enjoyed my flights on JAL and ANA within Japan. I have not yet tried any of the low cost carriers though. And I get seeing friends, I usually have to plan my week ends to be in Tokyo or Osaka because that is where my friends are.
So now onto the itinerary!
Outside of Onuma and Hakodate I've not spent time on Hokkaido and so will leave that to other posters.
1) I can't remember what the Japanese phrase that is being shortened and my Japanese sucks but Hyperdia's "Seat Fee" is not the fee to reserve a seat. The cost of a shinkansen to from Tokyo to Kanazawa is 14180 with a seat and 13850 yen without a reserved seat during low season which I believe May post GW is. The Kagayaki shinkansen are all reserve seat and you will need a standing ticket if you don't pay to reserve a seat.
2) Once you're paying for a car rental I would just rent it from Kanazawa. I am unsure of what the expressway charges are for your trip but I know there is a CEP Pass: https://hayatabi.c-nexco.co.jp/page/?id=73
you could see if it saves you money. I have only driven in Tohoku. I just paid out of pocket for the expressway one day (~2000 yen total) and got a TEP Pass for the consecutive days of travel.
3) This is the bus ticket you want: https://option-ticket.mitsuboshi-kaidou.com/mitsuboshi-pamphlet_en.pdf
While it says it is only valid until March 2020, a variation of this ticket has been around for quite some time. It use to be offered in Japanese only and had slightly fewer stop options and a shorter time frame, but all the places you want are in both passes and I think even the shorter time frame won't affect you. The one available to Japanese residents is 6000 yen.
You take the bus: Kanazawa to Shirakawa Go (day 1 of Pass)
Shirakawa Go to Takayama (day 2)
Takayama to Hirayu (transfer there to Kamikochi and pay out of pocket for a round trip night in Kamikochi or Hirayu) Day 3
Hirayu to Matsumoto (day 4)
This keeps most of your travel days relatively short. Kamikochi is lovely but expensive (cheaper options include a cabin but finding dinner and breakfast can be problematic.) I've only day tripped to Hirayu but the onsen were great and I know @LikeBike enjoyed staying there.
I'm not a fan of Shirakawa Go. I thought Hida no Sato in Takayama was more interesting. But it's on your way to Takayama so it makes sense to go there.
I took the shinkansen from Nagoya to Tokyo and then transferred to Maihama last trip. I just waited in line at the ticket window because I planned to charge my tickets. It was pretty easy.
Flying within Japan is pretty easy.