I presume you are talking about bringing your dog along in the cabin, not in the hold. Did you check AC's size and weight limits (27*40*55 cm, 10 kgs) and does your dog+carrier meet those?

Even if there were seats available on the nonstop AC flight, you could not take your dog with you in the cabin. AC uses a three-class 77W on the YYZ-NRT nonstop. In-cabin pets are not allowed on these aircraft in J (and PE), presumably because of the way the suites/seats are laid out.
Even service animals are restricted to guide dogs only in these cabins with the restriction that "the animal travels under the Suite's footstool and/or seat at all times" - which will effectively prevent you from putting the seat in bed mode unless you have a perfectly adapted dog who's fine under the seat like my lady . That negates much of the reason to pay the hefty premium over economy.

As this will be your dogs' first flight - I would only consider taking the nonstop flight if you are absolutely sure she'd be OK with staying in the carrier uninterrupted with no need "to go" for the 14 hours it will take from getting through security in YYZ unless you get outside or to the animal inspection facility in NRT.
Personally, I'd not go via ORD on a connecting itinerary for the same reason but connect in YVR or SEA to shorten the TPAC leg with an animal on her first flight.
Transiting the US with a dog is uncomplicated. If you have the paperwork to get her into Japan you're all set.

NH does not allow in-cabin pets at all on your itinerary - you'd have to check her as baggage. See NH pet policy.

UA's cabin pet rules are similar to ACs (carrier size 28*28*46cm, no weight limit though). UA does not allow pets as checked baggage, you'd have to fly her cargo using "PetSafe", which is cumbersome and very expensive in comparison.

As far as age goes - my dog is ten years old as well, and flying with her is absolutely uncomplicated.