Good questions. The "fat man" and "switch" cases you described have been discussed ad nauseum in philosophy and more recently in psychology. These discussions have focused almost exclusively on the different intentions of the person deciding whether to act so as to kill one and save five. In "fat man" the person intends to harm and kill the one as a means to save the five (saving the five is also intended), whereas in "switch" the person intends to save the five knowing, but not intending, to kill the one as a side-effect.

I think this difference would in fact be significant in a court of law, where I doubt prosecutors would press charges in the case of "switch" (and I doubt juries or judges would convict--of what crime exactly?), but they likely would in "fat man" (e.g., assault and battery, perhaps manslaughter). I actually think the law would be getting things roughly right here. An important part of law is preventing risky and dangerous behavior. In "switch" there is little risk of harming more people than by doing nothing. In "fat man" it is far from clear that the person is justified (as much as he'd need to be) in believing he will save five rather than risking killing one as well as five dying. And in any case, we don't want to encourage people to risk killing other people when they think it might save more people (oops--but what about war?)

Alas, I've tried to test whether people's moral intuitions are being properly influenced by their epistemic intuitions (about whether the person is justified in believing their action will work), and the results did not come out the way I'd hoped (but they raised some other interesting questions). But your question raises another under-discussed confound in the cases. Typically we think that it is morally impermissible to disobey a law (at least a just law). So, if people are asked whether pushing is morally permissible in "fat man", there is one more reason to say 'yes'--that it is likely illegal--than in "switch".

This is an important and difficult question. If we answer in the affirmative, then suspects are likely to know that they may be lied to by law enforcement agents. Still, they may nonetheless often be fooled or tripped up (they don't know when an officer is lying and when she is being truthful), and this in turn could lead to more convictions of guilty people which in turn would reduce recidivism and increase deterrence, thereby reducing the victimization of citizens by criminals.

I would think that a practice of telling lies that may be helpful for finding the truth (e.g., by eliciting a full confession) is justifiable if the reduction in crime it engenders is sufficiently large -- and this is quite large. Such a practice should be carefully circumscribed and supervised to minimize harm and to suppress abuse. And the lies should be revealed to those who end up not being charged -- and revealed also to those who will be charged, and before their trial.

Circumscription is important. Lies should not be very painful (e.g., telling a suspect falsely that his beloved mother has died) nor risk provoking a false confession (e.g., telling a suspect falsely that his buddy has implicated him and that he can save himself from the death penalty only by admitting to the crime in question). An example of a permissible lie would be telling a suspect falsely that a certain witness to what happened has survived and will be able to testify.

I don't think there is a hard-and-fast rule to give here. Do you call the cops when you see your kid litter? Of course not! Just make them pick it up and give them a good lecture about why that is unacceptable behavior. But if you see them commit murder? Well, yes, then it seems appropriate. If I caught one of my children shoplifting, I would try to come up with a way to make them repay the store--but I don't think I would be supportive if people at the store gave me an indication that they aggressively prosecute every case of shoplifting.

I think our responsibilities change in different relationships. I would also try to "correct" minor misdemeanors (like littering) when done by friends or more distant family members. The worse the crime, the more it seems to me to call for a legal report. But I think we are, in a way, much more responsible for the behavior of our minor children than we are after they have reached the age of majority, and we are much less responsible for distant relatives, acquaintances, and the like. So my own culpability in failing to report some law-breaking is relative to the degree of my responsibility for the behavior of that other person.

Do I call in every case I see of someone speeding past me on the freeway? No. That's not my responsibility. But if I see evidence that they are seriously impaired in some way (weaving dangerously, etc.), well, yes, I would call that in.

I think the only good advice I have to give here, beyond such rules of thumb, is that you exercise the best judgment of your own level of responsibility (to the criminal, to his or her victims, and to your fellow citizens) and of what you can do that is most likely to provide the best available resolution to the situation.