How should you interact with a police officer - what are your obligations, your rights, and how should you conduct yourself?

I'm a law student. I'll take this one. This applies to the US specifically, though being polite and deferent are probably universal.

In short: TL;DR answer: Be polite, calm, and friendly. If you are guilty of a crime, admit nothing, do not give permission to search anything that would be incriminating, say that you don't want to talk to the officer (unless answering extremely general questions), and, if you are detained, ask to speak with a lawyer. Be more compliant if you are innocent, but if you get the slightest hint that they think you're responsible, stop complying and ask for a lawyer if detained. For more mundane interaction (i.e. speeding tickets) be polite and deferent, and don't confess to anything unless they totally have you nailed. Arguing with cops will very rarely advance your case; save that for court if you care enough to challenge the ticket. More detail follows.

In minor cases (e.g. speeding tickets), you generally want to be polite, deferential, and honest, but probably don't volunteer too much information, except insofar as it's obvious. If you were going 85 and the cop asks why he pulled you over, it's probably wiser to admit you were speeding than to play stupid; in some borderline cases, being honest and likable will get you out of a ticket or into a lesser ticket. Arguing with police officers is generally not going to get you anywhere. If they're wrong about some material fact, you'll probably have to deal with it in court. Being calm, friendly, and deferential (address them as "officer") is often your best chance of avoiding a ticket, and will almost always avoid any escalation. In some cases, crying or explaining yourself may work, but if they don't believe you, it may make things worse. Similarly, if you made some mistake (i.e. did not see the speed limit change) it may be helpful to say as much politely, but again, you won't win an argument.

For more serious offenses (basically, anything criminal greater than a speeding ticket).

Edited to add: Basically, never talk to the police or other similar authorities under any circumstances, except where it can't be avoided, e.g. speeding tickets.

A police officer is either detaining you or they are not. If they are not detaining you, you are free to stop talking with them and leave. If they are detaining you, you have the right to remain silent and the right to an attorney. If you are being detained, and you ask for an attorney, ALL QUESTIONING MUST CEASE. Anytime you hear a story about some guy the police were grilling for eight hours: if he'd asked to speak with an attorney, they'd have had to stop.

In general, if you even think you might be guilty of something, it is best not to try to explain yourself and not to make up excuses. Most criminals don't think they did anything morally wrong. The police will not share your perspective. Especially if you are guilty, you should ask if you are free to go, and if you are not, ask for an attorney. This is advisable even if you are innocent if the crime is significant.

The police CAN legally lie to you in order to exact a confession; this is a rather common tactic. That means they can tell you someone has positively ID'd you, or tell you that your fingerprints have been found, or that your accomplice has turned on you even when these things aren't true.

Of course, if you actually have an accomplice, you should hope you've both credibly committed to cooperating in a prisoner's dilemma. Omega cannot save you now.

You should never give police permission to search anything unless you know that there is nothing incriminating there. If the officer tells you that the law entitles him to do something, and then ask for his permission, you should probably tell him that he does not have your permission, but if what he says about the law is true, you're not going to stop him. Even if the police find incriminating evidence, if they did not have a legal right to search where they were searching (i.e. they lack probable cause), that evidence generally cannot be used against you in criminal proceedings.

If police are questioning you about someone else (who is not a spouse) who may have been involved in a crime, it gets fuzzier. I'm not entirely sure how extensive police power is; ultimately, the state has some capacity to compel your testimony (there's no right not to incriminate others), but this generally doesn't work because someone who doesn't want to testify can generally testify to a lack of memory on whatever issue (as people might do if threatened by the mob).

It's also worth noting that roommates and people living with you can, under certain circumstances, authorize searches of your possessions. They can certainly authorize searches of common areas.

This is endlessly more complicated, but this should be a pretty good overview. You cannot be compelled to say anything incriminating, and if the cops are bargaining with you, that probably means they don't have enough to get you on. Again, if you've done something, or if they think you've done something, you're going to want a lawyer to sort things out. The risk is obviously a lot higher if you're guilty, but you can run into serious risks even if you just seem possibly guilty.