Is it permissible to wipe over the two socks after pouring water on them until it reaches the skin, knowing that they were worn in a state of ritual purity, and what is the ruling on previous prayers performed in this condition?

Firstly: The fundamental principle regarding socks and water is purity. They are not to be judged as impure merely based on doubt. Therefore, do not concern yourself with searching for an impurity unless you are certain of it.

Secondly: Water reaching the skin of the foot while purifying the sock does no harm. Wiping over the sock is valid if it was put on after a complete purification. Jurists have differed regarding the condition of the khuff (leather slipper) preventing water [from reaching the foot]; the most correct view is that it is not a condition, as there is no evidence to prohibit wiping over it as long as it is considered a sock and people customarily wear it.