How much does your dog weigh? Korean has a 7kg weight limit now (they raised it from 5 about two years ago), TK has an 8 kg limit.
Both airports have transit hotels, but both hotels do not allow pets. The one in Istanbul is notoriously difficult to deal with even with service dogs which in theory are allowed.
I`ve always entered Korea during Incheon layovers; the requirements are straightforward (microchip, health certificate, rabies titer test) and the inspection process quick and easy.
If you do not want to leave the airport for the layover, routing via Korea with its day-long layover is not convenient at all as you'll have to spend a full day in the transit area with no designated pet relief area. This will not be fun.
Ten hours in Istanbul is manageable in comparison. IST has a pet relief room in the transit area.
To my knowledge, the 14-day quarantine requirement still applies for travelers with pets entering Turkey, so unless that has been lifted it's not an option to leave the airport if traveling via IST.
Personally, IŽd see that the requirements for the dog to enter Korea are complied with, fly with Korean via ICN and spend the day-long layover outside the airport in a hotel. I can recommend this one - inexpensive, nearby, pets allowed, and there's parks and grassy areas nearby for nice walks.
If you want to stay in the airport, route via Istanbul.

You'll need to get to a CFIA office and have the paperwork processed there. Call the nearest office: https://inspection.canada.ca/about-c.../1313256130232
I have no idea where you can get your dog tested for Covid, but I guess the CFIA will be able to tell you that as well.