1. Make sure you can play the purchased music using iTunes. On your computer. Just load up iTunes and try to play the music. If it plays, great. If not, then not as great. It should ask you to put in your username and password if it needs to authorize. After it authorizes and starts playing, then you're good.
2. Using iTunes, put at least one purchased song onto the iPod. You MUST use iTunes to do this. What this will do is to create a file on the iPod called "iSCInfo" or "iSCInfo2". This file contains the keys to allow the iPod to actually play your music, and only iTunes can put this file there. Without this file, purchased music will not play on the iPod.
3. Use Media Center or whatever program you like to put those M4P's onto the iPod now. They should play just fine.
As for your son, if he's syncing using iTunes then he needs to do step 1 and make sure that iTunes itself can play the songs. If iTunes can't play them, then the iPod won't be able to either. There's other issues that can cause song skipping as well though.. If he has a 4G iPod running firmware 3.0.2, then it can corrupt files when it goes into deep sleep mode (after 36 hours of inactivity) and this can cause it to be unable to play purchased music until you resync with iTunes. Right now, the only fix for that is to go back to firmware 3.0.1. There's other possibilities too.