Sounds like you did get a pretty good deal. I've had problems finding compatible RAM for eMachines in the past, but maybe you've got better luck than I.
I heard that if the HDD is bad then it would make a noise or something. Is this true? Yes, it can make clicking/grinding noises when it's going bad. That's about the time you want to save what you can off your system and go shopping for a new hard drive.
And if the CPU is bad, more than likely the tower won't boot. Yes, most of the time, CPUs either work or they don't.
If the power supply is bad then it won't boot up. If the power supply is dead, it won't boot up at all. If it is failing, you may not get power to all devices on your system or it may be intermittent.
We are looking into the RAM because it is the cheapest to replace and is usually a major cause of stop errors. We'll start here and if we need to continue on to other items, we can address them later.