Is it possible to remove and replace this register duct boot without screwing up the dry wall around it?
Large Images: https://imgur.com/a/lh9HIt2
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Very often the boot simply rests against the back side of the drywall. The sheet metal flange receives the register screws, and the drywall is pinched between the boot flange and register when the register screws are tightened.
You could prove this out by pushing upward on the register; if the register is connected to flexible duct it will easily push upward and away. If there is rigid duct, or it there is insulation pressing down on the boot, it may not move or may be difficult to move. You may be able to remove it from below (if it is flexible duct and there is enough slack) by pushing it up and turning/manipulating it. In my house, the duct is larger than the hole, so the boot cannot be pulled down from below.