What kind of method is used to repair a hole like this?
Also wondering if there could be any health risks being near the hole? (e.g. harmful substances in the insulation)
You need to cut/saw out a square to remove all the damaged drywall, screw in blocks on the back, get some drywall, cut and fit the piece to the hole, and screw it in.
Then joint compound using paper strips around the seams and smooth it out for painting/wallpapering.
You can tape over some plastic if you won't fix it in the near future to avoid getting dust and dirt into the house, but I don't see it as particularly dangerous😊