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I'd guess around half of the PBP games currently running on this site are Paizo APs or Modules, so that's definitely ok. The biggest copyright concern would be maps and artwork; you shouldn't simply use those as-is. Making your own versions based on the originals is fine, of course.
Adventure Paths are ready-to-play; each AP is a series of 6 large modules designed to take the PCs from 1st to 14th-18th level.