A ceiling fixture may be installed one of two ways, it may have screws that hold the base to the mounting strap on the outlet box to support the weight, or a threaded rod and coupler may be used to suspend the fixture. To install a fixture with screws, locate the base and make sure there are two holes that line up with the screw holes in the mounting strap. Also check that the screws to be used thread freely into the holes before holding the light up to begin installation.
There will probably be several options for an electrical source to serve a new outlet. For instance, it is likely there are several suitable wall receptacles in any given room to extend the circuit to another room or another floor of the house.
Reinstall the wallboard strip. Trim along the edges with a sharp drywall knife first to remove any torn paper that could created a gap behind the molding.
Mark each stud for notching using one of the nail guards as a template. Use a handsaw to cut the top and bottom lines for the notch to the depth of the plate thickness.