Yes, if your fan stops working because your power supply went bad, the CPU could overheat very easily and shut down or even crash resulting in loss of data.
Computers can have more than one fan in the case. The purpose of the fan(s) are to circulate air, drawing air in from the outside and blowing it through the power supply, over the CPU and other heat generating components to remove the heat they generate and carry the heat outside the computer box or cabinet. Without this, the heat buildup in the computer can cause components to overheat and fail leaving you with a computer that does not work.
CPUs get power from two places-- From the motherboard through the tiny pins on the bottom of the CPU (Which in turn gets power from the 20 / 24 pin connector that connects tot he power supply) and from a 4 / 8 (Only on the newest boards) - pin connector that directly feeds the CPU a very steady supply of power. Both come from your actual Power Supply.
You can customize your computer in a process called "overclocking" whereby you make certain BIOS and/or hardware changes to increase CPU voltage/current settings, or timer latch settings, and this will make the CPU "clock" faster. When a CPU runs more cycles per second, it also consumes more power per second, meaning that it also dissipates more power as heat per second, increasing its temperature. If this is not properly managed, the CPU could overheat, which can permanently damage it.
Neither, The CPU is the Central Processing Unit, it only processes information and bits of data the computer sends thru it. The motherboard supplies power to the case fan AND the CPU fan, which is supplied power from the PSU (power supply unit)
Pulling the plug isn't too bad... it's re-inserting it that could cause problems. It could cause a power surge.
Yes, almost every PSU has enough power to supply a P4 CPU. It depends on the total load, not only on the CPU.
Many/most motherboards have a separate power connector for the CPU. Check your motherboard's installation instructions to locate the CPU power connector. Then connect the matching power supply cable to the CPU power connector on the motherboard.
I know it as a P4 power connector. it works for many CPU's.
Pull out CPU and turn on your computer, if you hear BIOS error complaining about CPU absence: most likely your CPU is dead. Also try to test power supply harness. All required wiring diagrams and standards you can find in internet, just use google.
No
A CPU works off of DC voltage. If you monitor the current draw of a CPU, it will likely not be DC, since a modern CPU is a CMOS design, current is only drawn when transistors are switched (ignoring the small leakage current). This is why overclocking (forcing the transistors to switch faster) can cause the CPU to overheat, and why "power save" mode on a laptop sometimes slows down the processor speed.
A power supply that transforms AC to DC is used to provide the proper voltage to a CPU.