Each disc installs the new hoods and new templates in the program files on the machine, and they stay there. If you have BG, BV, and AL, but not OFB, Seasons, or Freetime, you'll have the three original neighborhoods, the three vacation destinations, and Belladonna Cove, but you won't have Riverblossom Hills and Desiderata Valley.
Andrew, I sometimes find that a sim who can't be ordered to clean up dirty dishes can be influenced to do so. But it can take a long time as for some reason nobody seems to have fixed the code that makes non-controlled cleaners clean up absolutely everything else before taking care of the dishes!
I've had this glitch more than once. Once everyone in the house had it at the same time and I finally resorted to the stuck object remover.