I will answer your question with another question. If you do not know how to use the box to install a receptacle what makes you think you are qualified to do electrical work? I understand homeowners doing their own plumbing, if you screw up there is a leak, or you can smell something is wrong. If you screw up in electrical work you will kill someone or start a fire. I hope the $60 you save by moving your outlet by your self does not cost someone their life.

In short, no you can’t, screwing through the box would indeed be a violation. It will NOT pass inspection unless its listed for use in sush a way. . NFPA 70E states it has to be listed for use, a nail can pass through within 1/4″ of the back of the box if its listed etc etc.. even that may have been recently changed I don’t recall exactly. Most old work boxes are not. Some are made with 2 screws inside them already I think there called smart boxes, but you have to watch your wire fill with those. Anyhow get a nail-on box there less than a dollar

You are defeating the purpose of the old work box, they are strong enough on the sheet rock for general purposes and the receptacle will not take any more pressure. But it’s up to you.

no that would be fine so long as you don’t leave the screw protruding where it would contact the wires.