I know in my desktop I have Office versions 2007 and 2010. I think it installs into it’s own directory structure under “Microsoft Office”.
We have Office from 2000 all the way to 2010.
You can install office 2003 on a machine that previously had 2007, but it will probably not recognize it as an ‘upgrade’ in which case you should just remove 2007 and then install 2003. You will also then need the compatibility pack so that 2003 can open any documents created in 07 or later.
Hope this helps!