So default on the server is to stop it 15m after someone logs off. But if I want to run it all the time so that my friends don't have to restart it before joining, will time still pass when no one is on the server? As in, will it be day 50 even if no one has played that entire time? And dinos will still eat and mate and all that?

As long as the server is up and running then Yes time will continue to pass with or without players online. Shutting it down is the only way to stop the passage of time in-game.