How can I know if I damaged any electrical wire while drilling a hole in my wall?
The strict answer to your question is that you can't, practically speaking, at least without a visual inspection. You'd need a boroscope or access to the area. Partial severing of a conductor would not disable the circuit, but it could cause a hot spot in cases of high, sustained current flow.

Wiring is not run level with the outlet boxes. It's typically 8-12" above them, with drops into the boxes from above. The cables might also run along the studs from above or below. 



However, wiring is supposed to be bored through and mounted to studs at their depth center. Screws 1-5/8" long should not contact them.