When do I use spackle vs. more sophisticated materials? How big does the hole need to be before I call in a pro?
Well, a proper drywall fix for a hole will require the use of drywall mud, spackle, patches, or any other gimmick that is offered to homeowner at the local store is a product that will not yield a professional result. You state that you have a 3" x 2" GAP in the wall. Did you mean a hole? In any case Drywall is something that can be done by a homeowner. Yes, it can be messy. Yes, there is a certain amount of skill that can be obtained to help facilitate the work. But I find that the Skill helps you work faster. You can be patient and get a pro result on your own. Youtube is a really good tool for you here. But if you want to get this wrapped up quickly, then call a drywall guy. It is a small patch. Usually a drywall guy will charge $300-$350 as a minimum to show up and do small work like that. Go around the house and see if there are other holes you want patched. Make that $300 count.
Use a drywall jab saw, or razor, to make the area a perfect rectangle.
Take some 1X2 pieces of scrap wood , about 5" long, angle it in the hole, then fasten some screws through the existing drywall to the 1X2- this becomes your backer. ( make sure to fasten both the top and bottom of the 1 X2.)
Cut a patch piece of rectangular drywall about 1/8" smaller in each direction and fasten it to your 1X2 backing.
Tape, mud, finish, sand the patch