When a research paper's term of copyright ends, say, N years after publication, does the publication house have to make it freely (i.e. no cost) available for use by general public? Assume that currently it's charging some money to download papers.
No, why should it? Running out of copyright explicitly means a loss of a right. It does not mean the beginning of any additional obligation.
Running out of copyright, however, means that anybody who already owns a copy may duplicate it and make it available under any conditions he chooses (including making it available free of charge). Many university libraries do this.