I will try to answer the best I can with the crazy late shift and lack of sleep.
1. What responsibility to the average Citizen does Tennessee LE have relative to preventing crime? I'm not responsible to the average citizen. I work for a city, the main welfare and protection as a whole is my goal. If I can protect one citizen daily good, if I can protect entire neighborhood great!
2. If someone is a victim of a crime more than likely they will not be entitled to any compensation from any LE department. But this is not for me to decide this is determined by a judge and jury. Typically you would have to show an officer to be extremely negligent and somehow purposely allowing someone to commit a crime that they know will endanger the public safety. Example I stop a drunk driver say screw it I'm tired allow him to drive off and half mile down the road he hits a family and kills someone. You bet that's probably gonna be a civil suit.
3. I don't know of any case law which would apply here. Most oaths of office never mention swearing to an individual citizen. I could be wrong on case law but never been shown to me.
4. I can't speak for LE as a whole or gather a general opinion about the 2nd. I commend citizens who enjoy their right to bear arms and do so legally. I don't always agree with firearms laws but that's above my pay grade.
These may not be the answers you were hoping to get but you pose general questions and some that are not the general scope of Law Enforcement.