I actually just grabbed a workstation, loaded MOSS 2007 on that, joined it right to our existing production AD domain.

no issues.. because it is its own application (Ie. it didnt modify AD schemas or anything like that that I am aware of)

You can test the v3 beta  in the same domain. It won't cause any problems.

One thing I forgot to mention though.... Our current
SharePoint domain is a Windows 2000 domain. Will this still work, or
do we need 2003?

We do have the beta  set up right now in its own domain - but
everything, including the domain controller, is on just one box. This
isn't a suggested configuration and we had to make changes to the
registry to get it to work at all. Because of this, I'm a little wary
of using it.... What do you think? Can we leave it as is, or is it
worth it to move it to our existing SharePoint domain?

depends on the extent of your testing...

if it is a separate forest then I would reinstall it into your production forest. That way the users will be able to access it easier with built-in authentication and it will be less work for you. Think of all the stuff you had to to do make it work on a DC. It will be similar if you have a completely separate forest, you will need to do xtra to get the "production" feel for proper testing.

The version of the domains do not matter to the best of my knowledge.

I can only suggest based upon my test. I installed MOSS 2007 only once on a workstation class machine (beefy tho.. ) in my production domain right at my desk. It worked great.