1) So the phrase "To Protect and To Serve" means nothing anymore? Maybe I'm misunderstanding, but are you saying that if a cop drives by a man on the street getting mugged at knife point, he could keep on driving by?
"To Protect and To Serve" is just the motto of the LAPD. Adam-12 and Dragnet got it national recognition. It has absolutely no legal content whatsoever. It means as much as most corporate mottoes.
The simple answer is that yes, the cop could drive on by. There is a long series of court cases. They are unanimous in their conclusions. Combine the doctrine of
Sovereign Immunity with the bedrock principle that the police are there to investigate crimes and act on warrants, not protect any individual citizen absent some sort of personal services contract. The law is very clear. They have no legal obligation to protect you. They may choose to or they may not. And when they do not no law is broken and a lawsuit against them will fail. It really is as simple as that.
There was a case here not too many years ago. Three police officers watched someone get knifed to death. They did not get involved. They took notes and arrested the suspect later. The lawsuit was quickly thrown out on precisely the grounds outlined above. By taking notes and investigating the crime after the fact they had discharged their legal obligation.
I've been to a number of police training events over the years. At every one of them at least one older presenter says, usually in almost exactly these words "First you go home. Then your partner goes home. Then the citizen goes home. Then the bad guy goes home." When it comes to keeping people alive you are #3. That's not necessarily a bad thing. The job is dangerous enough without taking extra risks when you think there's a good chance you'll get hurt. Pull back and call for backup is S.O.P. for many situations in departmental policy manuals. If someone dies (cf. Columbine) that's regrettable. It's not necessarily contrary to what the officers were told to do. And it's neither against the law nor grounds for a suit.
In the end you are the one who is responsible for your own safety. The police are not. Even if they wanted to be they couldn't do the job. US Presidents have the best personal security in the world. They still get shot at. Sometimes they get hit.
Criminals do bad things to people. When that happens to you most of your options are bad. They don't include things like "honor", "fairness" or time for calm, clear in depth reflection. That's why the legal tradition of self defense is the way it is with terms like "choice of evils" and "affirmative defense". Killing people is bad. It is less bad than letting innocent people be killed or maimed by violent criminals. It would normally be wrong to kill anyone, but in this case it's a perfect legal defense if the jury believes you.
None of the people in either of the cases we're talking about went wild, took the law into their own hands or did anything else wrong. They all acted well within the tradition of self defense against "the immediate and otherwise unavoidable danger of death or serious bodily injury." They did not do the police department's job. It's not the police department's job to prevent crimes from happening although it's a happy thing when it does. They did their own job according to the traditions of US and English Common Law. The kid with the samurai sword may have technically exceeded his authority once the burglar had left the house and was running away. But a case could be made in his defense.
In the pizza parlor case the employees would certainly have walked if the manager had cut the robber's throat like he was a kosher chicken. He had a gun and was wrestling for control of it. He had already pointed it at innocent people. A reasonable person in the manager's situation knowing what the manager knew could reasonably have concluded that innocent people were in immediate danger and that severe measures were the only safe way to stop the danger.
If the robber had been knocked unconscious and been tied up and they had gutted, cleaned, skinned and jointed him it would have been another matter. At that point he was not a danger. Further action was not reasonable self defense. Up until that point? They acted with extreme restraint by just cutting and whacking him a little and dragging him outside. Personally I would have at least restrained and possibly searched him and called the police. I wouldn't trust him not to have another gun and not to come back. Best to have him where the authorities can pick him up and take him away.