The mods can verify on a patching level, but I'd assume it should work. Adobe lets you use any account on at least two devices logged in simultaneously. The GenP guide only mentions the account should be in no way associated with any genuine Adobe subscriptions or your real identity/details. If you do encounter issues though, it's super easy to make and verify a brand new account for your new PC. And you'll have to patch the apps with the latest GenP version regardless after installing, no matter what account you use (again, as long as it's never been associated with any genuine subscriptions)

As already answered, it will run wherever you install Adobe and sign in with your free Adobe account.

"Adobe individual subscriptions allow you to install apps on multiple computers (such as a home desktop and a work laptop), but you are limited to having them activated (signed in) on only two computers at a time."

GenP only patches locally, so the Adobe limits don't count.