No, in other words, states and local governments are also bound by the Constitution in many of the the same ways that the federal government is.

The major difference is the Tenth Amendment, which sets the states apart by specifying that any powers not "delegated to" the federal government are reserved exclusively for the states. (In practice courts have found many "implied powers" that are not explicitly enumerated).

Federal laws are distinct from the Constitution.

No, those aren't other words for the GP's statement.

The Constitution, its Amendments, and decisions of the Supreme Court are not 'federal laws'.