My PS4 doesn't work with online games but Netflix works online

Sounds like you don't have an active PS+ account which is required for online gaming, whereas Netflix is third party and doesn't require an active PS+ account.

Physical game copies will have no issue playing on any PS4 you put them in, or else rental stores would not be able to rent out games.

You might run into issues playing the digital games (you mention these are "pre-owned") as those are licensed per the PS+ account that bought the games. To play these digital games the PS4 must be set to be the PS+ accounts primary system, or that account must be logged in and used to play the game.