In a word, No.
Though oversimplified, a quick and dirty way to think of it is: City officers don't have the authority to enforce county ordinances, even though their city lies within the county. Their jurisdiction (think "authority") flows from the city itself, and extends only to the city limits. They can, however, enforce state law within the city.
County officers cannot enforce city ordinances, even though the city lies within the county. Their authority flows from the county itself, and extends only throughout the county. They, too, though, can enforce state law throughout the county.
State police officers cannot enforce city or county ordinances, but can enforce state law anywhere within the state, including within city limits and in any county. Their authority flows from state law. (BTW, fish and game officers have statewide jurisdiction also, just like state police officers, and can enforce any state law anywhere within the state, not just fish & game laws. They are all peace officers who have attended POST, and have the same citation and arrest authority as the state police. Just a bit of trivia.)
Federal officers derive their authority from federal law, and can enforce federal law anywhere within the United States, including within cities and counties. City, county and state officers are not authorized by federal law to enforce federal laws.
But they all cooperate.
Some cities enter into joint power agreements with neighboring cities, or the county, to cooperatively enforce state law within their jurisdictions, so they can jointly patrol some areas and nab suspected violators outside their jurisdiction - though they usually call in an LEO from the appropriate jurisdiction to take over.
The doctrine of "hot pursuit" allows officers to chase suspected violators across city, county and state lines (though radio is faster, and neighboring jurisdictions or states are usually contacted in advance so they can set up roadblocks, spike strips, block traffic, and/or join in the chase.)
Federal officers sometimes collaborate with state, county and city officers, sharing information and federal resources to assist state and local officers in investigations, and once the investigation is complete and arrests are made, they divvy up the charges based on whatever was found. If a state case looks shaky but there is evidence of a violation of federal law, the matter would likely get prosecuted at the federal level. Sometimes a person can be prosecuted both by state authorities and by federal authorities for the same offense - think of a drug dealer being caught with 50 pounds of MJ: he can be prosecuted under state law for possessing it, and under federal law for bringing it into the state.
Clear as mud? If so, sorry - just trying to help -