Problems installing Snow Leopard (10.6) on Mini

Question

I did a clean install on my Mac Mini (Macmini3.1 - Intel Core Duo, 2 GHz).  It came with Leopard (10.5), which I installed, including all the updates.  However, I got an error message after installing part of the Snow Leopard (10.6) DVD that was burned by the store where I bought the used mini.  The error message was that the disk needed to be cleaned. I used a DVD disk cleaner, and blew some compressed air into the DVD slot.  I also wiped off the DVD with a lint-free cloth. However, now it just ejects the DVD every time.  I have installed using this disk before with no problem.  In fact, this is only the third time the DVD has been used.
I've tried the DVD in my MacBook and it mounts with no problem.
I ran hardware test, and it shows no problems found.
I've inserted other Apple software DVDs, such as iWork and iLife, and it rejects them as well.  However, it never rejects the original install DVDs.
I tried using a retail version of Snow Leopard, but it immediately ejects it. 
Do you think the optical drive is failing, or have you got other ideas what might be causing this?

Answer 1

Thanks for your suggestion, Arch_PL, but I could not get any farther than seeing the DVD icon, selecting it and pressing return, in your point #5 above.  After that, the Mini remained in permanent startup mode, i.e., grey Apple icon and spinning beach ball. 
My MacBook has 400 firewire and my Mini has 800, but I do use an adapter.  I do not know if that contributed to any problems.
The Apple Support article, HT1661, explains that some Intel-based Macs may need a firmware update if the target mode does not work properly.  However, I looked at the firmware update article, HT1237, and it does not appear I need any updates.