From my understanding you don't need to give your password to get iPhone screen repaired, is this correct? And can you wait there?

While they fix it?

Added (1). Also what are the chances of them stealing information from my phone despite not having my passcode?

No they don't need your password, and no i don't advise waiting unless they tell you it's getting fixed right away. Most companies have a line and your phone may be behind several others