What is a valid proof of deregistration, and why do I need it?

I am checking out soon and need to show valid proof of deregistration. What does this mean?

When will the deposit be refunded?

I had my Check-out 5 days ago, but I did not receive my deposit yet.

What happens to my mail that arrives at the accommodation when I've already checked out?

I have already checked out, but what happens to my mail?

Can I check out earlier or later than the end date of the tenancy contract?

My check-out is planned for 31 August, but I have a flight on 23 August. Can I check out before 31 August?

At what date and time is the check-out scheduled?

The end of my contract is near, but I don't know when my check-out will be scheduled.

Is a pre-check-out mandatory?

I am checking out anytime soon. Do I need to do a pre-check-out?

At what date and time is the pre-check-out scheduled?

I am going to check out in 2 months. What is a pre-check-out, and when is it scheduled?