Problems with POSTing is usually (though not always) related to the motherboard. If you have another MB or PSU to test with, you should try that. You mention video card power problem beep sequence, my guess would be it's the PSU, but as I said, get a different MB or PSU to test with. Maybe you can borrow it from someone, if you don't have extras yourself.
My initial thought is that this sounds like some classic symptoms of bad caps. Given the age of that board, I wouldn't put it past being at the point some caps are failing.
If my thought is correct, then the problem will only get worse over time. The upside is that if you're a crack hand with a soldering iron, or know someone who is, it's a relatively inexpensive thing to fix if you can find some capacitors of the same specification. The downside is that if you're NOT a crack hand with a soldering iron, and try this yourself, you'll probably just make things worse. So in that latter case, it might just be time to consider either retiring that system or trying to find another board to replace it.