Is the District of Columbia a state? Does it operate like a state? Does the District of Columbia have a govenor? Is it special in the way it operates?
The District is not a state, and does not function as one.
The District does not elect any Senators, nor any Representatives. It does elect one "Delegate" who sits in the House, and may speak in debate, but does not have the right to vote.
The Constitution allows congress to exercise "Exclusive legislation 8in all cases whatsoever" over the District. However, since 1973 the Congress has allowed limited "home rule" in the district, although it could cancel this at any time by a new law.
The district has a local government, but it more closely resembles the government of a major city than that of a state. It is led by a Mayor and a city council.
It is specified in some Federal laws, and one Constutiuonal provision that for certain limited purposes the Districtm should be treated "as if it were", a state.
There have been a number of proposals to make the District into a state. The most recent ones have proposed to do this by redefining the "Capital District" into the White House, the Capital building, the Mall between them, buildings adjacent to the mall, and a small adjacent area. The rest of the district would, under such proposals, be admitted as a new state. Congress has the power to admit new states, so this would not require a Constitutional Amendment. But no such proposal has yet passed Congress.
There are some special provisions in the way the District is Governed. For one thing, to compensate for the many public building and facilities (which pay no taxes) and the special demands on the District of the Federal Government, Congress provides some of the funds needed to run the district, with the rest coming from local taxes. Congress must approve the District Budget, and can veto local laws.
See the Wikipedia article for more detail.