If I read a math proof in a paper that requires payment to read, is it legal to rewrite the proof with different words, phrasing, and order, then post the proof publicly in a website (with a citation)? Or we are forbidden from publicly sharing math proofs until the paper's copyright expires?
A proof can be protected by copyright. The underlying facts of math cannot. But if one has copied details of the order of the proof, or of the selection of theorems to use, and if several other choices would have been possible, then the new proof may constitute a trivially modified copy, or a derivative work, and in either case making of it might be copyright infringement.
However, making and distributing a copy, even with no changes at all, for purposes of comment and criticism, might be fair use in the US, fair dealing in the UK or some other parts of the Commonwealth, or fall under an exception to copyright in other countries (these generally vary significantly by country). This is usually a very fact-driven question.
Copyright law only protects "original expression". The originality of a mathematical proof lies in the underlying idea, but that idea is not itself protected by copyright. What is protected is the expression of the idea. Typically, however, a mathematical proof contains no original expression, it contains a standard symbolization of that idea. The surrounding text in ordinary (typically minimal) could be protected, except for formulaic utterances like "Therefore:", "It follows that"... As the US Copyright office says, mathematical principles are not protected by copyright. Facts are not protected, and every mathematicians that I have known considers their proofs to be facts.