So before I continue on to answer your questions, I have to say I have my GF's box running on your PSU with the following specs:
Athlon 64 X2 89w @ 2.5ghz
Visiontek HD3870 512MB
3GB of Ram
Gigabyte 785G Motherboard
3 HDD's and 1 CD drive
1 Wireless Ethernet Card
Its been running for about 6 months now and has has zero issues, I've also stress-tested it for 8 hours and nothing overheated or cut the power out. In-case you don't know the HD3870 draws far far more power than either of the cards your looking at. You'll be fine with a 4770, so I highly recommend that option. Also, compared to the 4670, it has the new 40nm die shrink, which means its going to run cooler all around.
So after that...onto your questions:How do you adjust the speed of the fan on the graphics card? Is there a program?
There are many programs that will do what your asking. Many times the Graphics Drivers will allow you to do this natively. On an ATI card, this would be called the Catalyst Control Center, than you would navigate to whats called "ATI Overdrive" and manually move the fan speed there. Another option you have is MSI Afterburner (for those of you who don't know, this is based of riva-tuner but is fully compatible w/ ATI now), this program has a great read-out and is very user friendly for changing fan speeds, clock speeds, etc. Also, you can have Afterburner change the fan speed depending on what the current temp is. That is very useful for low-maintenance builds. Another program would be Riva-Tuner.How do you check the temperature of the Rig and Card? Thermometer?
The things mentioned above will also to Temps for you. All of them will, and will be more accurate than an external Thermometer (it has built in sensors). If your worried about heat the 4770 has a dual-slot cooler. That will help keep it nice and cool, as the hot air isn't pumped back into the case but is pumped out of it. Way more efficient.How do you calculate the power consumption of the computer?
There are some power-supply calculator's floating around the internet that you could use, but really I would not recommend them. You can go to your local home-depot and pick up a watt-meter for about $6, or one off newegg called a kill-a-watt. http://www.newegg.com/Product/Product.a ... 20a%20watt
These will show you what the draw is from the wall, which is IMO the important thing, though if you wanted to figure out what the actual draw was, you would use the efficiency rating of your PSU and than factor that into the electricity being drawn from the wall.What is the best way to push a computer to the point of frying itself? And how would I avoid this?
There are many programs for this. CPU wise we are looking at OCCP, Prime95, Intel Burn-Test, Hyper-Pi, and Linx. Any of those will push your CPU to a heavy load/heat/power consumption point. I would recommend them in that order, though Prime95 is probably the easiest to use. As for graphics, you could download ATI-Tool, which has a rendering stress test that does a pretty good job presuming you don't have a stupidly powerful graphics card. For testing memory/overall stability use memtest, and fill it out to use your appropriate amount of ram. I need to mention that all of these can be found through a quick google search and are free! Presumably to test the system under stress you would run all these tests at once, and you are pretty much assured a good figure of how stable it is.
Avoiding damage to your computer is simple. Keep it dust free, and good ventilation. Most systems, including yours, should be fine with the cooling they came with. If these stress tests don't raise your CPU temp above 60C, or your GPU above 85C, you are good to go. To monitor temps of all hardware at once you can use a program called Hardware-Monitor. Its not as exact with readings and utility features, but it gives a good overall display of your temps all at once on the same page.
I hope I've been some help, goodluck and tell us what you decide on! Last, but not least...
Welcome to TR!