I'm willing to bet your problem is a GPU failure. Have you tried booting into Safe Mode (hold shift at startup until the progress bar completes)? Most of the white screen/black screen/lines in screen problems I've seen have been GPU failures.
When people bring me iMacs with GPU failures, what I usually do is pull the GPU out, hit it with 175C heat for 10 minutes, and then replace it. I have yet to have an iMac GPU *not* revive itself via this method. Then I tell the customer that their problem is a GPU failure, and I explain to them that I cannot fix the problem because the only proper way to do that is to replace the GPU with a brand new chip (which are basically no longer available). All the replacement parts for sale are junk GPUs sold by people who have done exactly what I did except they're lying to their customers by telling them they are working chips. I inform the customer that their iMac is now working, but that the problem is not fixed, and that they should back everything up while they can and just use it til it dies again (as to when that is, I tell them it could be anywhere from a day to months). I only charge them the diagnostic fee because I haven't fixed the problem.
If you're willing to pull it apart and have a decent hot air station, then you can try this procedure to correct the GPU issue, but note that this isn't an actual fix, this is just a temporary remedy.