So the product key from a retail Windows 7 disc should activate a Windows 10 upgrade, installation, or reinstallation after repairing (replacing) a motherboard. Theoretically. Maybe only for the first year.
Does Windows 10 offer you a place to key in a product key from an earlier version, when you install it from scratch, or need to activate it again after a repair?
You must perform an upgrade from either win7 or 8 to generate a product key which gets stored in the cloud. No other way around it. You can't boot off a DVD.
After that you are then free to boot off a DVD and format your OS partition if you want to.
If you want to get the actual product key afterwards (already in the cloud) there are some third party utils or registry hacks. Haven't done that I will do later (even though it may be a red herring).