You've got a very old (2005), second hand Mac which you indicated is wearing out (e.g. screen starting to go) plus second hand DVD boot disks that sound like they are incompatible. (You said you "copied the DVD from work" -- that does not sound like a likely to succeed scenario; also, running Disk Utility from an incompatible boot DVD is like playing with fire.) If they are incompatible, you will NEVER have success in booting from them, and thus might never be able to resolve your hard drive issues, which indicate either hardware broken or a faulty OS install.
The only way to sort this out is to get a set of RETAIL (generic, not machine-specific) 10.5 DVD boot disks (or disk) in good shape. This will likely cost $150 or even more. You may still find you have an unusable computer, but it's hard to tell at this point. If you can boot from the CORRECT DVD, you will then either be able to archive or erase and install a new 10.5 OS, or you may discover hardware problems remain that make the machine unusable.
This is probably not what you want to hear, but if I were you I would take the mahcine to a shop where someone can take the hard drive out and put it into a USB or firewire enclosure. Then you should purchase the best Mac you can afford WITH its original DVD set and hopefully you can recover your files from that drive when you connect it. I think this 7-yr old Mac will never be a reliable device for you in the future.
If you really want to try a Hail Mary, at this point I can only suggest:
(1) Replace the PRAM battery. This should cost less than $10. (I doubt this will solve your issues, but it might)
(2) Read this article http://support.apple.com/kb/TS1417 and try booting into Safe Mode. If that does not help, try booting into single user mode and run the "fsck -fy" command as explained in the above article.
Any serious hardware problem will probably cost more to repair than getting a new Mac and finding the parts will be a real hurdle. Plus they'll just be old, used parts not likely to last very long anyway.
Sorry you're experiencing this. I have a 2005 G5 2 GHz iMac with 10.5 now and it's somehow running just fine, but I bought it new and have been very careful about keeping it clean and up to date. These can be fun machines, these iMacs, and they're historical as well, but a 7-yr old Mac can go at any time. I keep it backed up to both Time Machine and cloned to a separate external drive just in case. I used to have a 2001 iMac which stopped working in 2009 after 8 good years, and I even had a 1994 PowerMac 7100/66 which I recycled in 2005 even though it was still working!