Restretching one straight edge of carpet should not be a major challenge and unless the carpet is old and subject to tearing is probably a do it yourself project. You will gain a lot of respect for carpet installers kicking just one wall length in place though. You will need to rent the tools. Your other option is to slip an installer doing a job near you a few buck when in the area.
Adding a layer of drywall may not accomplish what you hope I fear and you will be adding unnecessary weight. No big deal so long as wiring is long enough but your switches and outlets will need extender boxes.
You are going to gain the most sound insulation by putting something in the wall. You could rent a machine and blow in some cellulose or fiberglass.
To be honest, I would make or buy some foam (fire rated) sound baffles before layering up with dry wall. You can be clever and work them into the decor with colored fabric or even art or photographs digitally printed on fabric. Google decorative soundproofing for ideas. Here is just one I found quickly: