Is it permissible to wipe over one's shoes after taking them off and putting them on again while keeping the socks on, and is it a condition that the shoes cover the ankles for wiping over them to be permissible?

It is permissible to wipe over a sock worn with a sandal (i.e., a sandal worn over it) under two conditions: First, the sock must cover the obligatory area (the feet), and it is not a condition for the sandal to cover it. Second, the sandal must not be removed after the invalidation of the purity (wudu') in which the feet were washed. If the sandal is removed and then put back on after the ablution in which the sock was wiped over, the purity becomes void. A condition for the permissibility of wiping is that the sock should not have a tear that prevents continuous walking. It is permissible to wipe over it as long as its name remains (as a sock) and walking in it is possible.