I guess more specifically, when cops order you to sit down in an interrogation room (before you have been charged or arrested), is this a 'lawful order'?
I am asking because of the way this is often shown on shows like Law and Order. It seems like this is just so the cops can bully the suspect and be able to dominate them, and so is there a legal basis allowing them to do that?
If the police order you to sit in an interrogation room and you are not permitted to leave, by definition, you have been arrested. Detaining someone against their will for longer than necessary to answer a few questions on the spot (which is a lesser imposition on your freedom called a "Terry stop") is what it means to be arrested. Legally, the police are only allowed to arrest you if they have probable cause to believe that you have committed a crime.
I believe that you are confusing arrested (being detained by law enforcement against your will for more than a Terry stop), with being booked, or being charged with a crime. Generally speaking people are arrested first, and then booked next, and then charged with a crime after that, although this isn't always the order in which this happens.
In much the same way, if the police observe someone committing a crime, they will first handcuff them which places them under arrest, and then book them sometime not too long later when they arrive at the police station, and then formally charge them sometime after that after a conversation with the prosecuting attorney to see if the prosecuting attorney is willing to pursue the case. Booking generally involves bringing someone to a police station, getting identifying information, taking a mug shot, taking finger prints, and depending upon what the police want to do, searching your person and inventorying your possessions prior to putting you in a jail cell in jail garb.
Usually, however, the grand jury or the prosecuting attorney (it varies by jurisdiction) does have subpoena power to compel you to provide information under oath prior to trial as a witness, following the service of a subpoena upon you a reasonable time in advance as set forth in the relevant court rules, unless you invoke your 5th Amendment right against self-incrimination and are not granted immunity from prosecution based upon your testimony in exchange, or you invoke some other legal privilege against having to testify.
They can, of course, simply ask you to come to an interrogation room and answer questions, and merely imply that it is mandatory without actually saying that you must and without clarifying the situation. In that case, which is extremely common, their legal right to interrogate you flows from your own consent. If you answer their questions, your answers could provide the police with probable cause to arrest you that they didn't have when they started asking questions.
Indeed, often, when police interrogate you before booking you, they are doing so because they need your statements to establish the probable cause needed to legally arrest you. This is why criminal defense attorneys counsel people to immediate ask for a lawyer and refuse to answer any questions other than those needed to establish your identity.
You can also ask if you are under arrest and if you are free to leave (which are mutually exclusive).
If they say you are not under arrest, you are free to leave, unless you are appearing pursuant to a subpoena.
