I have purchased and downloaded the 4 GB Mountain Lion update. Can I burn this to a DVD and share it around with my family? Is it locked to my Apple ID or anything?
Yes, you can burn a DVD from it. The instructions are the same as for OS X Lion and need to be performed before you upgrade (otherwise you don't have the Install OS X Mountain Lion application anymore).
If you don't have the download anymore, you can probably get it from somewhere else; since you bought it this could be legal depending on your jurisdiction. The
When installing OS X ML from scratch, it (repeatedly) asks for your Apple ID, but entering yours can be skipped in all of these dialogs.
Note that the following restrictions apply (license agreement PDF, emphasis mine):
This is in contrast to the section A above it, restricting the license of a preinstalled Mountain Lion to the single computer it came bundled with. So in some situations, you are allowed to install one ML purchase on multiple computers.
So it could be legal to install it to other computers in your home you control (e.g. your children's), but it's prohibited to share it more widely.