No you do not need to replace the insulation, it needs to be fluffed back up a bit. On the left side it is compressed, just grab the outside and pull it a bit, at the most, reach inside, behind the visible face of insulation and fluff up behind it, or pull it out, reshape it to fill the cavity. There is no special magic there to insulation, it needs to fill the cavity, that's it.
In my opinion, since it is inside, the OSB does not need to be repaired. In most jurisdictions, only one wall covering is needed by code. If it is structural, as in wall bracing, since the whole wall is covered in OSB that hole will be of no concern. If you wish to fill it, you only need to add either a 3/4" or 1/1/2" (2X) scrap that laps halfway behind the right hand vertical edge and screw it in place with 1 1/4" screws, the kind you will need for the drywall too. If the left side is cut flush the framing add a 2X block to create an attachment point for the OSB, screw it in place with 2 1/2" screws. 2, maybe 3 screws will suffice for these attachments. If the repair guys did nice, they cut the piece out so it was salvageable, set it back in after all old fasteners are removed. Screw it in place with the 1 1/4" drywall screws. If it was not salvageable, cut a new piece to fit that is 1/4" smaller than the hole, 1/8" on each side, and screw that in place. All these dimensions are rough, no need to be exact, the main thing is to get it in flush, with not too large of gaps.
Remove all the old drywall fasteners, with pliers or claw hammer. Determine if the sheetrock is 1/2" or 5/8" thick. Cut you new material 1/4" smaller than the hole you are filing, and screw that in place with the 1 1/4" screws. Remove the wall plate and notch for that too. 5 or so screws across the top and bottom, and 4 down the ends and center will hold it in place. Apply the drywall joint compound, NOT spackle tape and smooth out to match existing finish, add a second coat after the tape coat dries. Do be aware you can press down on the taping knife fairly well to bed the tape in, keeping the knife clean of excess mud and the wall from a heavy build up at this time. Then when that dries, add the second coat if you choose too.
The drywall looks like it is only taped, not second coated. This is called "fire taping" when it is the only step done. Additional steps are "joint finishing"