To go along with what nigel said, here's an analogy. Picture dropping a pebble in a pond, and it sends waves outward in all directions. Now you can imagine that if you drop another pebble somewhere else in the right way, you can cancel the waves in a particular target spot. However, look at the rest of the pond; now there's two sets of waves everywhere, so the "noise" is twice as bad. The reason noise-cancelling headphones work so well is they're very close to your ear, and they're in a shell and insulated enough that they don't produce very much noise to the outside world. They're focused on your ears, and nothing else.
Now picture throwing a big handful of pebbles all over the place, you end up with a big noisy mess. The most important point here is that there is NO POSSIBLE WAY for you to cancel all that noise by throwing a single pebble in a single spot. Just like there is NO WAY to cancel all the noise of a computer by placing a single speaker inside it.
Now notice that the only way you can cancel the waves of one pebble in all directions is to drop another one in exactly the same spot at the right time. In terms of sound, this is even worse, as you're in 3 dimensions now; You would have to have a speaker right on top of every noise source. But, of course, you can't place a speaker INSIDE a hard drive or processor fan or anything like that. So you would have to place it as close as possible. How close is "close enough"? Well at low frequencies (hundreds of Hz), sound wavelengths are on the order of feet, but at 1Khz the wavelength is just over a foot, so placing a speaker an inch or two away is a pretty significant error. So really, you will have to limit yourself to low frequency cancellation. Plus you've got the issue of your noise not being a point source. And, you have to consider how large of a speaker/woofer you would have to use to deal with a particular noise: a tiny speaker is NOT going to produce very low-frequency sound. And you also have to consider the directionality of the speaker, vs. the more omnidirectional nature of many noise sources.
My guess is that by the end of your experimentation, you'll end up with a case full of little speakers that sounds like a screwed up jet engine. Based on your questions, you must not know much about this... but all you need to do to get an "upside down" wave as you ask, is use an inverting amplifier... or even a non-inverting amplifier, and just flip the speaker terminals. That probably sounds way too simple to work, (and there's a good reason for that) but it's exactly what you're asking for.
Once again... there is a REASON you don't see this implemented in computers already. If noise cancellation was as simple as a single microphone, speaker, and a small amplifier circuit, then don't you think EVERY noisy piece of equipment would come with one preinstalled?