The rally last December was one of nearly a dozen paid-for protests organized by Russian émigrés in the U.S. in the past two years. They spent $150,000 to $200,000 in some months, accounting records indicate, to rally thousands of demonstrators near spots such as United Nations headquarters and the World Trade Center site. State-controlled Russian television, whose content is closely guided by Kremlin handlers, covered some of the events, often as the only news organ present, showing video of them on the evening news back home. Boris Barshevsky at a pay-for-protest rally in Queens, N.Y., last year. Organizers said the...