Tiger and OS9 use the same Filesystem HFS+.
Both of them are able to handle files with resource fork
and are able to show the right file type.
If the file type disapears
then there is something wrong with the filesystem
or the resource fork has been lost while copying the file.
Where are your files?
If they are at the Mac's hard disk
you should try to check your HD with hard disk utility.
If your files are at a server volume (Helios, Xinet)
you should use Helios or Xinet utility to rebuild the desktop file.
If someone used raw unix commands to move the files
the resource fork is lost and can fix them only the hard way.
Use the Finder to move files otherwise you have to be very careful
and choose the right tools!