The first thing to understand is that there is a difference between a codec and a container format. AVI and MOV are containers. They can contain many types of video streams in them. There is no telling what codec you need to decode the contents of some random AVI or MOV. But MPG and DIVX are formats. You need the MPEG-1 codec (which is included in practically every desktop OS) or MPEG-2 codec to decode MPG and a DIVX codec to decode DIVX video (which is often contained in an AVI container.)
As to determining whether or not DIVX is installed, just look in the Add/Remove programs listing in your control panel and see if it is there.
Honestly though, there is a big cesspool of MPEG-4 decoders right now: DiVX, XviD, 3ivx, etc...
This is a big mess and if you're not looking to create videos, then the best solution IMO is to avoid it and just get the excellent and free ffdshow decoder which decodes all XviD, DIVX, 3ivx, etc with a minimum of fuss.