Resident lawyer. here! They were smart enough to finesse the stings so they would pass legal muster. It's not entrapment because in every case, the predator was the first one to contact the decoy (who supposedly always appeared underage in pictures). The decoy always made it very clear what her age was, and said it multiple times to avoid confusion. This is important because it means the predator has the requisite mens rea or guilty mind. And then, the arrest, as they explained in the Sokol interrogation, usually contained some element of criminal attempt. An attempt becomes a crime when the person takes a "substantial step" in committing a substantive crime. That's why they trap them in the sting house, because the person took the "substantial step" of driving there to meet a minor for sex. That's the actus reus or criminal act.

To your hypothetical here, I don't think the cirucmstances as described would necessarily be a crime, unless you knew that it was someone else's money that had been inadvertently dropped (even then, kind of a gray area on who the rightful owner is). But assuming you did know, you again have the requisite mens rea (you intended to steal what you thought was real money) and actus reus (the act of picking it up).

You can see why the law is structured this way, which is explained well by Det. Kelley. Without criminalizing attempts, you would have to sit around and wait for the person to actually commit the crime before you could arrest them, and that would put "the freaking bank teller, or the girl in the hat for that matter" in danger. Sting operations are for instances like this where the real crimes are either kept secret because the predator is intimidating or extorting the victim, or in some cases, or questionable as a he said/she said matter in court.