Do all PhD programs have publishing requirements? At the bottom of this webpage are publication requirements for PhD students at Skolkovo Institute of Technology. Is it common for PhD programs to have specific publication requirements like this?
I think the answer is no, not all do. But if you rephrase it slightly you get a different result. If we define PhD strictly as a research doctorate (though some schools might be weak on that), then a dissertation of some sort is required. In all fields, and for all advisors, I think that the attitude, if not the specific rule, is that the dissertation be of publishable quality. The exceptions, where the committee doesn't have such an attitude, seems to me to be a vanishingly small set. 

That is a judgement made by the advisor and committee in many fields. In some fields, and apparently at some universities, prior publication is required. But that is just a way of having an independent judgement about publishable quality; by the journal/conference reviewers. The decision is, then, somewhat distributed. 

Some dissertations never get published other than informally (university libraries, University Microfilms, etc). Some result, after the fact, in one or more publications. In my own case (though long long ago) there was no formal requirement other than a submission to what was then University Microfilms. But a shorter version of the dissertation was published a few years later.