Best JR pass combo

I have an itinerary in Japan for 10 days (after ending a tour of Kanto/Kansai we are spending some days on our own) and I am trying to figure out the best JR combo to get. Any help with the best JR passes and itinerary suggestions would be greatly appreciated!

Day 1: Osaka to Kyoto: leaving early in morning
Day 2: Kyoto to Takayama (pick up rental car) to Shirakawago
Day 3: Shirakawago to Takayama (return rental)
Day 4: Takayama to Gero Onsen to Takayama
Day 5: Takayama to Kanazawa
Day 6 l: Kanazawa
Day 7: Kanazawa to Tokyo to Hakone
staying in ryokan and need to get around locally in Hakone with buses/rail
Day 8: Hakone to Tokyo
Day 9: Tokyo
Day 10: Tokyo to Narita Airport

I calculated each leg you plan, and for Days 2-8 you do get some decent savings for a 7-day JR Pass.
Be aware though that Shirakawago doesn't take a whole day - you can see the place in just a few hours. Be sure not to miss the mountaintop overlook over the village.
https://youtu.be/lvtACNNuLUs

Takayama has some sights as well that are worth your attention. It's a bit of a toss up if you also want to see the Hida Folk Village there if you are also seeing Shirakawago. But it is much easier to walk in and out of a lot of old structures, and there is a lot better English explanation.
Kanazawa though is a lovely city and has some wonderful preserved areas. The crown jewel though is definitely the Kenrokuen Garden.
https://youtu.be/riB-vzWNKLM

As for Hakone, you can take the shinkansen up to Odawara, but then you'll need to pay for the rest. A Hakone Free Pass can be a good option if you are doing the Loop Tour. Be aware also if you want to hike through Owakudani, there is a 500 yen charge and you need advance reservations.

Thanks for your input, I was planning on getting the 7 day JR pass but not sure what to do for the last few days with Tokyo/Hakone, there is the Tokyo Wide Pass that includes Narita Express train but will not help with getting around Hakone. Is it better to just pay for individual fares in Hakone with cash or Suica card? We are a family of 4.

As I said, you can use the pass Days 2-8, which would cover up to returning back from Odawara for Hakone. The last 2 days, you don't travel enough to cover the costs of a Tokyo Wide Pass.
As for going to Narita, you can take the N'Ex, Keisei Skyliner, the cheaper Keisei trains instead, or from Tokyo Stn there is the Access Narita Bus for 1300 per person (but limit of one large suitcase per passenger).

Don't need a JR pass.

Redo the itinerary so you don't backtrack twice.

Kyoto to Kanazawa
Kanazawa to Takayam via Shirakawago using car
Takayama to Gero Onsen
Gero Onsen to Hakone via Nagoya

Japan Rail Pass does not sound best.

Takayama, Shirakawa-go and Kanazawa are connected by bus;
you don't have to rent a car or drive by yourself.
https://www.japan-guide.com/bus/shirakawago.html

H:
Redo the itinerary so you don't backtrack twice.

I agree.
I suggest you change the order of the places,
to save your time and cost.

When you don't turn back the way,
the JR regular cost for Days 2 to 8
(until you get to Tokyo)
is estimated as up to 22970 yen,
which is lower than the price of 7-day Ordinary-class JRP,
by several thousand yen.

JRP can cover your trip from Tokyo to Odawara and back.

On the condition that you use Hakone Freepass
and pay additionally the surcharge for Odakyu Romancecar rides,
the cost of your round trip from Shinjuku to Odawara
is calculated as 3100 yen (#1).
https://www.japan-guide.com/e/e2358_008.html

If you choose JR Tokaido Shinkansen over Odakyu Romancecar,
JRP is still worth considering;
otherwise, JR regular tickets seem more suitable.

(#1) The gap of the prices between
Hakone Freepass for the free usage area only
and
Hakone Freepass covering Shinjuku to Odawara round-trip basic fare
is 1100 yen;
the Romancecar surcharge is 1000 yen per way for this section.

----

Kyoto -- Gero -- Takayama (#2, #3) by JR within 3 days
: 4840 yen as basic fare.
Kyoto -- Gero on Limited Express Hida (#2, #3)
: 2750 to 3350 yen as Limited Express surcharge
for your ride with your seat reserved.

Kanazawa -- Tokyo by JR within 4 days
: 7480 as basic fare.
Kanazawa -- Tokyo on Shinkansen Kagayaki
: 6700 to 7300yen as Shinkansen surcharge
for your ride with your seat reserved.

((JR regular cost: 21770 to 22970 yen.))

Takayama -- Shirakawa-go by non-JR bus
: 2600 yen.
Shirakawa-go -- Kanazawa by non-JR bus
: 2000 yen.

Hakone Freepass (2-day)
covering Shinjuku to Odawara round-trip basic fare
: 6100 yen.
Shinjuku -- Odawara on Odakyu Romancecar
: 1000 yen as Limited Express surcharge.
Odawara -- Shinjuku on Odakyu Romancecar
: 1000 yen as Limited Express surcharge.

(#2) There is one Limited Express train
running daily from Osaka via Kyoto, Gifu and Gero to Takayama.
Limited Express Hida 25:
Kyoto (08:31 or 08:34, Track 0) -- (11:29) Gero;
Track 0 platform is near Central Gate.

(#3) Between Gero and Takayama,
trains are only very infrequently available,
and Limited Express trains do not go notably faster than Local trains.
Departure time of some Takayama - bound Local trains from Gero:
20:49 and 22:26 in the night,
07:35, 07:52 and 09:00 in the early morning.

----

With one or a few destinations
such as Gokayama and Kurobe Gorge,
Takayama-Hokuriku Area Tourist Pass
may work well and save your cost.
This tourist pass can cover the bus fares,
but cannot be used for Kyoto -- Gero -- Takayama direct route.
https://touristpass.jp/ja/takayama_hokuriku/