If you have Nero then you'll need the MPeg2 codec that comes with the newer versions of Nero. With that you should be able to burn dvd's. Likewise Nero will convert the AVI files over to MPeg2 which is the format required for DVD's. In the link I gave you there's also another program listed there (TMPG encoder) you can use to burn or prepare videos for burning to dvd. It is a shareware program however so you can only use it for a period of time before you'd need to register it. For earlier versions of Nero I think you need to purchase the MPeg2 codec separately from their site.

As for playback, QuickTime on its own cannot view all formats. You'd need to get the DivX codecs/plug-ins. At least with VLC all the codecs are built into the player itself.