If he has your wallet, merely, I hope you would not shoot him in the back. If he has your child, I really hope you would not shoot him in the back (or shoot anywhere in his general direction)!
I assume you're asking if I was a cop. Depends. Did I see him just commit a murder? Yep, I'm gunning his ass down. Rape? Yep, he's a dead man. Armed robbery? Yeah, but I aim low for his legs. Here's what I know for sure as a citizen. Let's say that a cop kills some small time thug in my neighborhood. What do I do? Well, I sure as hell won't burn down my neighborhood. I won't use it as an excuse to loot and steal from my neighbors. And I sure as hell won't say fuck it, think I'll go do a drive by. Seems the black community in Baltimore just doesn't get that.
I would shoot under both circumstances - and I am an expert marksman. But apart from that, I don't think I would shoot him, no.
Stealing a wallet is sort of old school. Lets pretend it is armed robbery at an ATM where I have just withdrawn money. A gun is stuck in my face and I relinquish my cash. The robber doesn't shoot me and starts to leave. Under those circumstances I would shoot the robber. My ability with a pistol is so poor that I would shoot at the center-of-mass, front or back. I would fire until the robber was incapacitated and unable to return fire. I would do this in any city in America regardless of gun laws. In many cities I would definitely face weapons charges for it. In those cities I might face manslaughter charges if the robber expired from his wounds.
I hope the police continue shooting people who attempt to get physical with them. Its called self-defense.
Depends on how they obtained the wallet. At least here in Florida. In many cases, the wallet is protected under current castle doctrine, wherein the thief's intention has already been established as hostile, and deadly force can be used against him.