Can you see WHO ran a background check on you?
I guess that depends on what sort of background check it is. Generally you have to sign consent for someone to do this. For financial, I have a service thru Bank of America called Privacy Assist and they email me whenever someone checks my credit. Keep in mind Bobby might be pulling Suzie's leg.
Background checks and name change?
Background checks are based on more than just name, including but not limited to date of birth, place of birth, address, etc. etc. More importantly your Social Security number, which you can't change. You can go down and file for a name change at any time through the state but your SS # will follow you forever, so your record will as well. Sorry.
Can a bad credit score interfere with becoming a police officer?
Yes it can, but her honest attempt to pay off her debt will work in her favor. She should just be honest with her investigator if the subject comes up.