Paulie was "a little sick all winter." I infer that Paulie was setting up a test for Sollozzo--checking to see if Fredo would drive for his old man, or somebody more formidable, and also seeing if the Don would leave the office at the same time even when Paulie wasn't there.
DC explained "going to the mattresses." In the film, Clemenza told Paulie to "order" the mattresses, not to transport them in his car. It was a ruse in any event, though the family eventually did go to the mattresses.