It sounds like you got a consular officer who was not trained and did not understand what she or he was supposed to do. Assuming the denial was...

You can check your case status online by going to https://egov.uscis.gov/ and inserting your receipt number.

Great! Sounds like they are ready to issue the visa. Stay in touch with them so you can receive it when they send it to you.

Your should go to the police and ask them to provide you with a police certificate. if they refuse, then you may not be able to get one.

Yes, even at the age of 18 you can write a letter to your grandmother inviting her to visit the U.S. You might want to add an I-134 Affidavit of...

In my opinion there is no such thing as an immigration Z hold. You may wish to meet with an attorney to explain what you are trying to find out.

If you claimed to be a U.S. national and not a U.S. citizen, then you may be able to argue you did not falsely claim U.S. citizenship.

It depends on how long she has been out. Generally she will be readmitted, even if they take away her green card and put her in removal proceedings.

They very well could deport him. He should file for adjustment of status immediately, and not wait for DHS to get him.

Perhaps there is an issue with payment. Or perhaps the attorney is not available. Perhaps the files are missing.