I'm new to these machines, not really a mac person, but I did recently pick one up, and have been reading up on them. Apparently the click/pop issue with screen blinking is a common issue with these, and it's generally attributed to the flyback transformer shorting out. Not saying there aren't capacitor issues - but that specific description - pops/clicks, accompanied by flicker on the screen, is the flyback transformer. From what I've read, it's a problem that can eventually escalate to the point that it's shorted out, and won't turn on anymore. That may have happened on yours. A way to test this would be to disconnect the display cable from the display port, and try powering it on with that disconnected. You won't have a display, but it should turn on and make the startup chime - if it's turning on with the display disconnected, but not with it connected, then that's probably your problem.
Also - my iMac is currently having occasional flyback pops, but separate from that it has an issue turning on at times. I think it's because the lithium ion battery on the board is dead, I've read that can make them act funny at times, replacement is on the way, we'll see if that fixes it. When it would first happen, I'd unplug/plug several times, disconnect the display - and no matter what, it wouldn't turn on. But after sitting for hours, overnight, it turns on again. Until it doesn't. But also, there's a PMU reset button on the board - pressing that once for a reset could clear an issue of it not turning on. I did run into the problem one day, and pressed the button, put it back together, plugged it in and hit the power button, turned right back on. I'm hoping the replacement battery will stabilize that, we'll see I suppose.