I downloaded a program in a .dmg format, but can't find anything to open (decompress) it with. I am running OS9.1 but also have 10.0.4 installed. Any suggestions will be appreciated.

.dmg is a OS X format file so i wouldn't recommend opening that file is OS9. Try opening it in OS X with stuffit or something.

I tried opening the .dmg program running OS 10.0.4, using Stuffit Expander.
I made a little progress. Now it tells me that the file does not appear to be compressed or encoded. Then proceeds to say that it appears that limewire may open the file - and would I like to try.

Have Disk Copy on OS X open the file.

Maybe I am missing something here. I tried to drag the file onto disk copy. Disk copy opened, but nothing else happened. Then I tried opening it from within disk copy, & the file was greyed out. I don't know the relevance that the icon is a piece of paper & the get info feature calls it a document. These are large (20-100MB) files, not a standard size document if you ask me.

use DiskCopy 6.4

I could not find disk copy 6.4. I already have 6.3.3 installed which was the newest one I could find other than Disk Copy X (which I have installed on my other OS- 10.0.4). That didn't work either.

Do yourself a really big favor and upgrade your os. Go at least 10.1.5 a free upgrade. OS 10.0 anything is no better than a beta, download an upgrade from apple, or use the software upgrade part of your os preferences. And just kind of informational, a dmg file should not, and as far as I know, can not, be opened in os9,(without additional software). It is a os10 disk copy file and usually contains os10 software.