Purpose of triangular placement of cones/spikes?

Primary reason is three points define a plane. If a fourth point is added and is not in the same plane (very likely unless the surface is perfectly flat) then it either does nothing or more likely makes matters worse by not having a solid contact with the object. Now if you have adjustable spikes/cones then you can compensate for the fourth point being in another plane. Someone else will most likely have additonal information as to the pros/cons of three versus four good contact points.

Sonically, I doubt that it makes much difference. I prefer four adjustible corner points mainly because it makes a more stable platform - less likely to tip over (speakers). But three points (adjustible or not) are much easier to level, as Pmotz suggested.

I think it is more important to decide where to put than how many. If the purpose of using cones is to "drain" vibration than I would like to drain them as soon as possible. That means they should go directly under the transformer and CD drive because these parts vibrate most.

A more scientific way is to use a stethoscope. Just put a scope against the bottom of the equipment at various points and listen for noise. Place a cone directly under the noisiest point and place the others evenly to balance out the weight.

Spikes under speaker are a different story. It is much easier to make fine placement adjustments to a speaker with three spikes than with four.

It's simply to insure that the feet or supports for the component all make contact with the shelf. If there are 4 feet and one is not making good contact, this can cause the chassis to vibrate, resonate. I think it's stupid to make these out of turned brass or other exotic materials. If you have a really cheesy rack, then soft rubber or sorbothane feet might help...