Hi Frank,
It was a shot in the dark. Well, since the problem 'followed' the install to another machine, it's not a computer hardware problem (unless you got identical machines), so, it must be
a. Your install media is not right (corrupted), or
b. The way you are answering the questions the installer asks you when you install Slackware.

Are you 'custom' installing, if so, try just using the 'default' answers using a 'normal' install.
Try a 'minimum' install (if there is such a thing in slackware, it's been a two years since I installed slackware).
Try installing Mandrake 9.1 or RH, and see if you get the same results. Maybe you have a friend with another set of install disks?
I'm just trying to give you some avenues to try, as I'm at the end of what I can think of. If none of these things makes sense to anybody, I'm open to any other ideas. I think my logic that the 'problem' follows the install disks is correct, therefore, it's an install disk problem or the answers to the install questions? Any takers?

You may try running knoppix. It's a debian based "distro" that boots straight off the cd (no copying anything to hard disk). It's autodetection is quite good (not to mention it boots off the device you're having problems with) so if it works just read off the fstab file there and copy that into your own fstab file. That may work.

Before Slackware, I used Mandrake 9.1 and I did not have issues with the cdrom drive...
On this computer, I used the recommended installation and on my old computer, I used the minimal installation. They both have this mounting problem... I don't think my installation cd is corrupted, because I surely wouldn't have been able to complete the installation.
Even if Mandrake and Redhat don't have this issue, I really would like to continue using Slackware