Thanks, Brett.
TrueView does indeed make the files readable in ACAD2000 again. It seems when I used ACAD2013 to view a folder, all DWGs in that folder became tagged as created in 2013, even though most of them had not been opened in ACAD 2013. However, once converted and saved in the Map home directory in ACAD2000 format as DWGs or DXFs, they are still not readable as GIS datasets in the Catalog; they still appear as single files. When I attempt to add the file, ArcMap says "Could not open the specified file."
I am running ACAD2000 in Windows XPMode (32 bit), under my Windows 7 (64 bit). Perhaps I should try to install XP in a separate partition and have a dual-boot machine. I hope I don't have to learn how to do that. I have 8G of memory, so it might work if that's what I have to do.