Is it permissible to wipe over socks made of cotton, polyester, and nylon that are opaque, water-permeable, and self-fastening on the foot? And is it permissible to wipe over two socks without removing the shoes?

The Shafi'is stipulated that for wiping to be permissible, water must not penetrate what is being wiped over. If water does penetrate it, then wiping over it is not permissible. This is the apparent (or prevailing) view of the madhhab (definition). The Hanbalis, on the other hand, do not stipulate this.

If the shoe worn over the sock covers the obligatory area [for wiping], then wiping over it is permissible, even if the sock beneath it would not be permissible to wipe over by itself. However, if the shoe does not cover the obligatory area, then wiping over it is not valid, especially if the sock alone is not sufficient for wiping over.