I'm sure this is going to go over familiar territory. We have a housefull of Macs and have had for a long time, and I'm fairly used to hunting down issues. This particular one is a problem I've fortunately not seen before. I suspect a serious hardware problem, but only got a look at it once it was fairly advanced.
My teen son was using this computer, a 21" iMac, I think from 2007 (it was a hand me down from his Mac-savvy Grandma). He says that it was freezing on him for the last week or two, and he'd have to restart. Then a couple of days ago when he restarted he got a folder icon with a ?, so he took the install discs and reinstalled the OS, which was 10.4. Yes, at this point he'd already goofed, because he'd gone over his data, and though he had an external HD, he had not used it with Time Machine for several months. I don't know what he was thinking, but he's 16...he probably was not.
This was where I finally came in. Yes, get old Dad involved once an issue has reached the crisis stage. :-\
At that point his desktop showed a fresh install of 10.4, which we wanted to update at least, and possibly retrieve his last Time Machine, but I figured there was likely an underlying problem. Disk utility showed that the disk needed repairing, but I figured that needed to be done from an install disk, and for good measure, I should probably run the Hardware Test as well.
Now things got complicated, because I have been unable to get the computer to start from any install disk I've tried. We have two sets of original install disks from this generation of iMacs (two similar computers), and complete install discs for 10.5 and 10.6, as well as Disk Doctor. The computer refuses to start up from any of them, with either 'C' or 'D' held down, nor will it start up in Safe Mode. And now the fresh install of 10.4 has disappeared as well.
All that occurs on restart, after the 'bong' is a grey screen. If a disk is in the optical drive, there is a brief sound of it spinning up, then silence for a good while, maybe 2 mins, then the disk is spit out. Nothing more. I have zapped the PRAM or whatever it's called these days, with no change.
We did have an external DVD drive somewhere and I'd like to try that, in case there's a problem with the optical drive itself, but I am thinking this computer may have both disk and hardware issues.
Any advice welcome...thanks in advance!