We have been moving out for almost 4 months now, and last week, we received a statement from them saying that we were still owing them more than 2000$ (1200$ were from the carpet). They said they did sent us the statements before, but none of us received any of those. There are 2 questions I need to ask:
1) Must landlord provide a proof of pictures of the damage for my carpet?
2) If all the tenants lived in the apt did not receive any statement within 14 days about this, do we have the right to ask about the proof from postal office showing that they've sent these before? And if yes, what would happen if they ignore our requirement?
Thank you for all of your help. I really appreciate it.

They need to provide proof . How long have you lived there? In some places there are wear and tear laws to prevent landlords from gouging tennents for normal upkeep.

We have been moving out for almost 4 months now, and last week, we received a statement from them saying that we were still owing them more than 2000$ (1200$ were from the carpet). They said they did sent us the statements before, but none of us received any of those. There are 2 questions I need to ask:
1) Must landlord provide a proof of pictures of the damage for my carpet?
2) If all the tenants lived in the apt did not receive any statement within 14 days about this, do we have the right to ask about the proof from postal office showing that they've sent these before? And if yes, what would happen if they ignore our requirement?
Thank you for all of your help. I really appreciate it.

I assume you didn't complete a walk through inspection with the landlord prior to your vacancy - it's a common, and often costly, mistake. Had you done an inspection, you would both have signed a sheet of paper that a)stated there was no damage to the apartment or b)a list of what damages had taken place during your occupancy.

The landlord will probably not be able to win a case against you without pictures of said damage - have they provided you with a receipt that shows the actual cost of replacing the carpeting?

I wouldn't bother asking the post office about anything. They're not going to remember if your prior landlord sent something to you or not. If you a filed a change of address at an appropriate time, then you would have gotten your mail forwarded to you.