I would think it's HIGHLY unlikely for your motherboard to die just by it touching the carpet. I've built on carpet numerous times, and sometimes putting the motherboard on the carpet because I got lazy to get the box
. But since I live in a relatively humid area, static is of no concern to me.
Also, it's recommended you build OFF of carpet, preferably somewhere without carpet and low chance of static (kitchen table, for example).
As for your PSU fan, some PSU's have a hybrid feature where the fan doesn't turn on unless it hits a certain temperature/load. Are you sure your PSU doesn't have this feature?
Anyway, have you finished building your rig? If so, did it POST?