The question that you pose can't be answered in the abstract and the conceptual framework you introduce in the section entitled "Resume" is largely unsound from a legal perspective, even if it may have some political theory usefulness.
Indeed, even the framework of "relative power" doesn't really work. The question almost always presents itself in a binary fashion. Either a law enforcement officer has authority to do some particular thing, or the officer doesn't. Frequently, there are multiple law enforcement officers in different agencies who all have the authority to do a particular thing, and whoever gets there first may do so.
In U.S. law, law enforcement is highly decentralized. There are hundreds of distinct federal law enforcement agencies, dozens of distinct law enforcement agencies in every single state, and almost every local government (sometimes even school districts, water boards, and park districts) have their own law enforcement agencies.
All told, the number of independent law enforcement agencies in the U.S. is in the low tens of thousands. Something on the order of 90%+ of those law enforcement officers report to local governments like cities and counties and have no direct chain of authority that goes up beyond that point. The remainder are split roughly equally between the state level and the federal level, but highly fragmented within each such level between different agencies within each state and with the federal government. The most common kind of law enforcement officer in the federal government is not the stereotypical F.B.I. agent. Instead, it is a federal park ranger. A few small states are more centralized (e.g. Hawaii), but massive decentralization is the norm.
The law enforcement officers in every single one of those agencies has a statutorily defined jurisdiction. Rather than being a neat hierarchy in which one law enforcement agent is always superior to another law enforcement agent, one has to analyze the circumstances in light of a particular grant of jurisdiction. There are general trends in how these grants of authority are made, but they are far from uniform.
Most grants of authority include both territorial and subject matter elements.
State police can typically arrest anyone for a violation of state law within their state, but not for violations of federal law or the laws of another state under circumstances where a citizens arrest is not authorized and where there isn't a warrant outstanding. A state game warden can arrest you for violating hunting laws in his geographic area, but not for violating state tax laws or federal immigration laws.
A village constable in a tiny hamlet in rural Tennessee can arrest the chief of police for the Tennessee Valley Authority's law enforcement agency (the TVA is a federally charted independent government agency), for drunk driving in that constable's village. In contrast, an F.B.I. special agent doesn't have the authority to make that arrest, but could arrest either the village constable or the TVA chief of police for taking a bribe contrary to federal anti-corruption laws.
There are law enforcement officers who specialize in investigating crimes by other law enforcement officers (often in the same agency). In state and local police agencies these are usually called "internal affairs officers", with a unit in the state police or state bureau of investigation that works closely with a unit of the state attorney general's office having a secondary role in this task.
In the federal government, the Inspector General's office in each cabinet department is typically charged with primary responsibility for these kinds of investigations. In the military, improprieties by active duty soldiers in connection with their implementation of the Uniform Code of Military Justice would usually be investigated by the criminal investigation service in each cabinet level department (i.e. the Army, the Navy (including the Marines Corps), and the Air Force) within the Department of Defense.
But in neither case is the conceptual issue one of authority. Usually any law enforcement officer in a particular agency has authority to arrest any other law enforcement officer from the same agency (with the possible exception of their direct superiors). It is a question of focus and job responsibilities, not of legal authority.
There is a notion of federal supremacy under the U.S. Constitution. In a bald showdown of authority between federal law enforcement officers and state law enforcement officers, say, over who is entitled to custody of a non-law enforcement officer criminal who has been arrested, the federal law enforcement officers will prevail.
But, these showdowns are rare, and interagency conflicts within state governments and the federal government respectively, if not reconciled sooner, are more often resolved by a Governor or President, through a chain of management command authority within that governmental entity, than in the courts. Even when there is no one shared chief executive between law enforcement officers, usually these disputes are resolved by negotiation between the chiefs of the respective officers, rather than through legal channels.
This is much less true in most countries other than the U.S. Most countries have far fewer independent law enforcement agencies, and have more clear and hierarchical lines of authority between them. This distinction is illustrated by your professor's example:
If the Queen of England pointed a gun at the President of America and
the President of America pointed a gun at the Queen of England, could
you prosecute and/ or charge any of them?
In the U.K., nobody has the authority to arrest the Queen, and the authority to arrest the U.S. President would be highly limited by diplomatic treaties relating to the treatment of foreign heads of state.
In the U.S., the question is tricker and quite fact specific.
The President has immunity for his official acts, but not for his private acts. If this standoff with guns breaks out on the floor of the U.N. in the context of a breakdown of negotiations, the President hasn't committed a U.S. crime.
On the other hand, suppose that the President and the Queen have been having a private tryst at a bed and breakfast at our small village hamlet in Tennessee where they have managed to escape their respective security details. (Pardon me for bringing such an icky vision into your head, this is purely for educational purposes and you should censor your own thoughts as you imagine it. I blame the questioner's professor for presenting a scenario that has this possible aspect to it.)
The village constable can arrest the President for this crime in violation of Tennessee law (assuming that the President doesn't have legal justification for his acts such as self-defense) since this act was conducted in an unofficial capacity. But he probably can't arrest and charge the Queen under diplomatic treaties that the U.S. has with the U.K. In contrast, a U.S. Secret Service agent would have authority to seize the Queen sufficiently to disarm her and prevent her from being a threat to the President, even if he couldn't charge her with any crime.