BLockOUT has some interesting questions, but answering them is not so simple.
There are different kind of capacitors.
By the nature of your questions, it appears you are specifically referring to electrolytic capacitors.
On top of that, there are also many different kinds of electrolytic capacitors, and each of them are best suited for a specific purpose. There are regular caps meant for general/bulk use and low ESR caps for high frequency use. In power supplies it means the general ones are for mains frequency stuff and low ESR ones for switching mode power supplies such as PC motherboard VRMs.
Any cap will eventually wear out, they cannot stand forever being used, their lifespan just is determined how much under stress they are, meaning temperature and ripple current.
But back to your questions:
1) Expected life of a motherboard is determined by the people who design it. If they are required to design a consumer motherboard, they won't design it to be operating after 20-30 years so they select components that are supposed to work the expected life of the product, whatever it is. Warranty period is a good sign of this, and in some countries there are legislations that a consumer product must be expected to have a certain life span and manufacturer is responsible even after warranty period.
2) Caps can look OK, and not work at all. They can be short circuited or open circuited or anything in between.
3) No, not really. If there is no proof the caps are bad then I won't change them just in case. I advise people not to recap everything just in case if they don't know what they are doing, because most often people don't know what capacitors are used for what purpose so which of them should be normal or low ESR or bipolar or something else. Some circuits NEED low ESR, some circuits are better with standard caps.
4) If they are there, they must be important for a purpose. If they weren't, they would be left out when manufactured to get more profits. But regarding whether they fail or not, usually smaller ones do not fail, but sometimes they fail too. These caps that are sprinkled all over for general purpose bypassing are not usually related to high ripple currents like those used for CPU or memory switching mode power supplies.
5) Electrolyte can dry so the capacitor loses its capacitance over time, yes.
6) That would be the low voltage bulk storage cap of the linear power supply, it is charged at 100/120Hz from the transformer and drained by the 12V regulator. Standard low frequency linear PSU stuff. if it ever fails it will most likely just lose capacitance and the 12V from the regulator would start become ripply before loosing regulation.
gdjacobs wrote:On some less expensive power supplies (for example, the infamous 250W Bestec models) the line for standby power is smoothed with a relatively small electrolytic capacitor and will go badly out of regulation when the capacitor fails.
But every regulator will go out of regulation when a capacitor fails, not just less expensive ones. And the capacitor may not be "relatively small" at all, because the higher the frequency of the switching mode power supply is, the less capacitance is required to go below the required voltage ripple specification. It might just be otherwise bad circuit design that stresses capacitors and all capacitors would eventually fail, or at least ones not rated for the given ripple current.