Display doesn't work – iMac does

Question

The display of my now 20 month old iMac behaves very strangely: sometimes when waking up from sleep or when (re-)booting the display remains black while the iMac continues working normally. The background lighting is not zapped (i.e. even if you look closely you cannot see a dim picture) – the screen is completely black, it seems as though the whole display is turned off.
After that and after rebooting over and over again (I don’t know if just waiting some time is enough – didn’t try that yet) the display will at some point turn back on, work normally and remain working – even for several hours – as long as I don’t turn it off. Then (always after I turned it off first, i.e. after shutting down or going to sleep) it may not work again. Sometimes the iMac will go to sleep and wake up normally (or even reboot), but never much more often then two or three times – then the display suddenly refuses to work once again.
When turning the iMac on it sometimes makes a faint if audible beep sound – not the speakers, but as it seems some hardware part (the power unit?!).
I already did a PRAM reset and a Hardware Test. The first didn’t work, the second threw an error at me: 4SNS/1/40000000:TAOP
This is very depressing for me: the display already was defect, luckily when I still had warranty, and was replaced. Then, one month after the warranty ran out the same display suddenly went black (just the background lighting, not the whole display), we let it repair (Apple refused to be tolerant and we had to pay 700 Euro for a new logic board), and now, only a few months later, this? It is very hard for me to accept that a computer has to be repaired three times in less than two years. The device was only lightly used and always stood at the same spot. I don’t get it. I will call Apple tomorrow and see what they have to say – maybe a miracle happens and they will be tolerant this time (I’m not exactly ready to pay hundreds of Euros again, especially if it’s always the display that’s causing the trouble) – but I do not have high hopes. I cannot explain this trouble I’m having with the display over and over again.

Answer 1

I've got my Mac back!
Even though the Finder wasn't running, because I had populated my Dock with dozens of applications, I was able to launch the Terminal app and open a window. From there I was eventually able to launch the Installer, point it at the 10.6.8 Combo Update (for some reason, fortunately, I had downloaded it rather than letting Software Update do it), and wait at least 30 minutes when the installer claimed that there was less than 1 minute left to complete the install. After it was done and I restarted the Mac, it no longer insisted that I needed to send/ignore thousands of "Finder quit unexpectedly." messages and the Finder was running.
BTW, towards the end of the installation (or re-installation), the Finder began to function (my desktop icons appeared and I could open folders) and now the only remaining issue is the Menu Bar won't appear unless I click the Finder in the Dock.
Thanks to those who helped me by providing the instructions needed to use the Terminal window correctly.