I can't speak for FCP or Logic, but there's no reason it can't run photoshop well. There have actually been threads in here with people stating how good it ran. Maybe some heavy filters on 4000k pixel images may be a bit taxing, but that's at the high end.
Basic web design is not taxing at all. It's just text editing, maybe a layout viewer like iWeb or Dreamweaver. The only thing that may cause strain is running a local database for testing, but MySQL or PostgreSQL aren't taxing, especially with a small DB.