After the rough in inspections that required open walls were complete we had the insulation installed. It only took one day. They covered several of the electrical outlets with insulation. We had insulation installed in the interior walls to reduce noise but it is probably more common to only have insulation in the exterior walls. With 2 x 6 wall construction we have a lot of insulation on our external walls.
To have good drywall you need good framing under it. Our framing was good and the walls look good. I have been told some people hide bad framing with thick drywall. They also hide bad drywall with wall paper. There is some special drywall used in showers and areas outside the house. We had some arched ceilings and a groin vault ceiling on the front porch which required a thin bendable drywall to cover it. Drywall cutting and sanding makes a huge mess. They managed to cover several electrical outlets and a window with drywall so you have to keep track of your outlets and make sure they are all still there after the drywall is in. The builder cut the drywall off of the outlets and fixed it. Again it helps to have pictures before the insulation and drywall so you can find things after they are covered.