To answer your final question, the 14th Amendment is usually considered to have brought state and local governments under the limitations imposed on government power in the Bill of Rights and other parts of the Constitution.
The original Founders would probably have agreed that states, and by extension cities, have such power. They were by no means libertarians. They were in favor of limitations on federal power as a means of protecting the governmental power of states, which prior to the Civil War implemented a great many laws we would find unconstitutional today. Including violations of freedom of speech, religion and press.
Whether the 14th was originally intended to bring states under constitutional limitations is a whole other question.
One thing I that we both can agree about without having previously discussed the matter, the drafters of the 14th amendment did not intend that the powers of the state would be limited by "emanations and penumbras" which only Supreme Court Justices could feel and see.
So far we have property rights, contract rights, due process rights, offered up by posters as reasons why the states are properly constitutionally prohibited from enacting rent control.
