and on the profile it said this account will be turning inactive in 0. Does that mean you can now use the name of one of their pets and their account is inactive or what?

No, "turning inactive in 0" means that the account will turn inactive in less than a month what can be anything between 1 and 30 days.

Also the first clearing isn't happening til friday 3 days from today so you may want to check back after 3pm subeta time to see if it has hit that stage where names can be taken

Thank you! :D