>>Who actually performs the background checks, is it a particular agency or is it the plant itself?<<
It depends. Some contract companies are approved to do their own investigations, yet others are not. And it can depend on the utility as to whether they accept an outside investigation. Some utilities have a contractor do all their background checks, yet some do it in-house. Too many variables to worry about or list here. Just know that it will be done and it will be thorough. They all know that the names you provide as references will shower you with praise, so they'll dig deeper looking for 2nd or 3rd layers beyond the names you provide.
A bad thing in your past isn't an automatic disqualifier. For example, I know of someone who spent a year in prison who has unescorted access. It depends on what the "bad" is.