It should be fine as the wavelengths are such that absorption will be nil. That said, you do not want to have a really deep pile actually blocking the port if the spikes sink into the carpet. If it is so deep as to do that, I would get longer spikes before adding any boards or other devices.

I have to think that the carpet is going to absorb a significant amount of low-frequency energy. This is especially true if it has a foam pad under it, which is fairly standard.

I would put a piece of wood about two feet square and an inch thick under the sub to act as a radiation plane. I think there is no need to screw it down, as the weight of the subwoofer is sufficient.

A few inches of carpet and foam will have negligible effect at subwoofer frequencies. Even huge traps do little below 100Hz.