I can play my purchased songs in iTunes but cannot import them into iMovie or iPhoto. Songs not purchased through iTunes import fine and purchased songs not marked "protected" are also fine
So, make a playlist of the purchased music you intend to use in iMovie, and burn it to a CD. Then, either use the tracks from the CD, or reimport the CD to iTunes and use those tracks instead of the originals.
I still have an external hard drive filled with iMovie projects that won't play the music
Won't play which music? Have you replaced the audio in those movies with the DRM-free track copies described above?