What authorization (if any) do crime investigators need to go around asking questions?

This is the part I seriously question. Nobody is forced to answer questions (or plead the fifth) under penalty of law except on the witness stand. If you decline to tell the police you know which way the perp went, or that you recognized him, you have not done anything that is a chargeable offense. If you were not a party to the offense or actively helping the person get away with it, you are not committing a crime.

If you deliberately mislead the police with false information, then you are committing a crime of obstruction of justice.

“Just asking questions” for no valid reason would lead to suspicions of harassment under the color of law, which is potentially a civil liberties issue.

OTOH, if there’s a legitimate investigation and John Doe was somehow connected to the crime (i.e. he knew the victim, he was one of the last people spotted in the area on surveillance photos, etc), well then there’s a legitimate reason for inquiry. If he didn’t do it, it’s unfortunate that he’s potentially implicated, but that sort of stuff does happen. One hopes that his name gets cleared and the police help him clear it

I should also mention that a police officer just going around sticking his nose into something without valid evidence that this could help them with a crime that they know has been - may have been, or may be - committed is at the very least theft of time, not doing your job. Most police are sufficiently busy that they don’t have time to wander around looking for unsolicited trouble.

About the closest I can imagine is police monitoring a stretch of road known for speeders, or an area of the city where theft is common. But then, this is monitoring a place, not surveilling a specific citizen with no indication the person has committed a crime. And - it involves a reasonable expectation that crime may occur in that location based on past history.

Basically. An investigatory detention needs to be appropriately brief and tailored to the crime suspected. In practice, it’s a remarkably murky area of the law.

I don’t think this is true. The police can investigate anything (or nothing) as they see fit. There are going to be internal controls to deal with officer misconduct and allocation of resources. But the legal restrictions are going to relate to the ability to compel or seize things – do they have grounds to detain someone, subpoena something, search something, etc.? In your scenario, the police would certainly be free to continue to investigate the threat (although probably unlikely to do so) using various methods – consensual questioning; surveillance; internal databases, etc. At some point, it could rise to the level of harassment or even civil rights violation, but not immediately.

I don’t know how common it really is to pick a target and then look for a crime. On some level, it’s a key component of a lot of white collar enforcement (you don’t know what they’ve done until you start examining things). You certainly get the sense that there are a number of high profile entities or individuals who attract investigatory attention for broadly political reasons, which is a significant burden to them whether or not a crime is ever identified.