Always, ''always'', have a USB boot disk to hand. It makes diagnosing problems much easier.
From my experience, a dead hard drive will result in a question mark being displayed on boot. Faulty RAM will involve a black screen and beeps. A failed display will give you the chimes but a black screen. On the logic board, there is an array of LEDs below the display connector, which will light up if the board is operational.
The white screen boot problem sounds like a bad graphics card. From my experience, the GPU is the most likely thing to overheat on an iMac. As you cannot easily open the machine, dust accumulates inside it, trapping airflow and eventually leading to overheating failure. Typical symptoms are random lines appearing across the display at all times, and / or booting to a blank screen. However, you can still usually boot to safe mode, even when the display is messed up. If you’re really stuck, you can hold down Control+S on startup and go into single user mode. If that works, your CPU, disk and RAM is probably okay - or at least okay enough to get a kernel initialised. You can then recover data from the disk.
Resealing the old card’s joints with a heat gun sometimes works, as does finding a replacement GPU from a donor Mac. For this model, it’s an ATI Radeon 2600XT. If you’ve already done a hard drive upgrade on a pre-2012 iMac, and fitted a CPU with thermal paste before, you should be able to swap out a graphics card in under two hours.
It’s somewhat ironic that given their reputation as “sealed” units, these old iMacs are some of the most upgradable!