I don't recall ever seeing a DVD for 10.3-all that I've seen have been the three CD sets.
In any case, what you've described sounds like the classic symptoms of a dirty optical drive. I've had many where the CD/DVD would mount on the desktop, but the computer would not boot off the disk.
There are special optical drive cleaning disks. The ones I use are an audio CD with a small, very fine brush attached to the surface. The audio disk will play and will instruct you to forward to specific tracks at different points. Doing as it instructs will bring the brush over the laser lens and clean it off.
If your iMac has Firewire and you have another Mac capable of running 10.3, you can always also boot the iMac in target disk mode and then use your other Mac to run the 10.3 installer.
Also, I would very strongly encourage you to consider upgrading to 10.4 rather than 10.3. The big advantage of 10.4 is that there is a current, up to date web browser for it-TenFourFox. If your iMac does NOT have Firewire ports, you will need to use a free program called Xpostfacto to install 10.4.