When recycling, the bolt rotates and goes backward to extract the casing. Does anyone know if the spent casing rotate together with the bolt?

It should. Ejector pushing on it and extractor pulling it. But maybe the case grabs the chamber more and doesn't rotate.

It may rotate or it may not, depending on how hard the case is wedded to the chamber walls at the time rotation starts.

When you see ejector and extractor swipes on the case head, the case is not rotating at the beginning of bolt rotation and the beginning of extraction.  In some situations it is from the round being over pressure and is stuck to the chamber wall.  Other times, especially with low pressure rounds like the .458 SOCOM, the timing is off and the bolt is rotating and trying to remove the case before pressures have dropped off enough to do so.  In this situation, the bolt is rotating but the case is not and the ejector and/or extractor causes a case head swipe that is mistakenly thought to be a sign of over pressure when it is really a timing issue, solved by a heavier buffer, stiffer buffer spring, or if you want to get fancy, and adjustable gas block.
 
So, when it is a sign of over pressure and when it it a sign of timing?

Look at the primers.  If the primer is flat as a pancake all the way across the primer pocket with no rounded edge left on the primer and the case head has swipes from the ejector and extractor, that's over pressure.  Primers are the softest part of the cartridge case and are the first to show signs of over pressure.  If you see swipes on the case head but the primers show no over pressure signs, likely it is not over pressure.
If the primer still has nice and rounded edges but the case head still has ejector and/or extractor swipes, it's a timing problem and you need a heavier buffer or heavier spring or turn down how much gas is flowing with an adjustable gas block.  All you want to do is delay the bolt opening a few micro-seconds for the pressure to fall off more before the bolt starts to rotate and try to extract the round.

So my AR is single shot (pull action). I did that by covering the gas port. Don't ask why as I live in NYS, and it is one of ways to achieve compliance. Anyway, would covering the gas port on the barrel cause over presure issues? Sometimes it is very difficult to extract the spent casing, and brass casings, in general, are easier to extract than steel casing.

Not overpressure due to blocking the gas port. That's because ARs have no primary extraction. .223 instead of 5.56 ammo should be easier. A longer cam should let you pull faster before it unlocks and yanks on the case. Look at UK rifles.

Primary extraction, Can you elaborate?

Yeah. I think that's what my problem is. Thanks.