Your ideas sound like good ones. Make sure you fill the gaps between the door frame and the studs - using insulation or spray foam.
You can buy special 'soundproofing' insulation from owens-corning. (I'm not sure what's different about it) If you don't buy that, get the highest R-value you can. Also, you might consider hanging sheet-styrofoam underneath the drywall (on both sides.) It's pretty cheap and is very effective sound insulation.