CD audio is indeed a device thing, like burning cds, or formatting drives.
In Slackware, devices already have good permissions. However, you need to add users to the correct groups. So if your user is foo, you need to do:
usermod -G cdrom,audio foo
Make sure you list all the supplementary groups you want your users to be in. I usually add floppy, too, so that I can format floppy disks from my user account.
The only complication is, that if you are using an IDE CD-ROM, your cd drive will be accessed through something like /dev/hdc, rather than a SCSI device. Now the group for all IDE devices is disk, not cdrom. Don't add users to the disk group, or they will be able format you hard drives. Instead, change the group of just the CD drive device to cdrom.
ls -l /dev/cdrom
To find out what /dev/cdrom is pointing to. It is just a symlink. If it is pointing to a scsi device (/dev/scd0 or /dev/sr0 perhaps) then everything is good. If it is an ide device, let's say /dev/hdc, then do:
chgrp cdrom /dev/hdc
If at any time you swap around your IDE disk drives, remember to change the /dev/hd? groups accordingly :-)
For your problem with audio even from root, michaelk's post is very good.