Difference Sheriff vs city vs state police?

That's interesting. I just realized that my "city" police are officially called "county police" on their website so that blurs police and sheriff jurisdictions more. I guess this is why there is an argument for abolishing the sheriff position here. Here, although the sheriff has some law enforcement abilities (I assume), I have never seen a sheriff pull anyone over or arrest anyone. I assume they can, but you don't see them patrol either. Even the only jail in county is located at the police station. Seems the position is more elected official then anything else. Maybe they do run the jail or act in the court room but I've never been in one so I don't know.

State police authority varies. In some states they are 'state police' and have full police powers. In other states, they are 'state patrol' and their primary duty is traffic enforcement. The state patrol does not take reports of criminal activity and does not investigate crimes.

Nothing prevents the state from working in city limits. They normally don't, but that is a matter of professional courtesy and not a matter of jurisdiction.

The Sheriff is the primary law enforcement officer for the county. The law allows that because he is elected by the people of that county. The sheriff is a manager, so it is not required for him to be a certified law enforcement officer in my state, but I'm sure it is a requirement in some other states. The sheriff is still necessary, even with state police, because he runs the county jail and is responsible for civil process service such as subpoenas, restraining orders, and eviction notices.

I can't see a sheriff's department being abolished. You may be the only city in the county, but I'm sure there are people that live outside the city limits. I've seen city police abolished and the sheriff's department take over, but never the other way around.

No and yes. While the rues vary with different cities and states, generally a city cop can cross the city limits and chase a suspect, serve a warrant or arrest someone. However it would be a arrest or warrant related to breaking a local law or ordinance. If it was breaking a state law, the local authority would yield to a state officer.

State troopers can operate within city limits. But their duty would be to enforce state laws, not local ordinances. They could, for example, arrest you for assault, robbery, murder, etc. But they would not be used to enforce a zoning law or evict a resident under a city ordinance.

THe same is true of federal agents. They usually only enforce federal laws.

However any agency can ask the others to help them enforce city, state or federal laws. The state could ask for local help or federal help if needed.

Because state A is not going to have officers in state B, if state A wants to serve a warrant on someone is state B, they would ask for state's B assistance in doing it.

But generally they can cross borders if in direct pursuit or if they have a warrant,