With regard to wiping over the socks, it is not essential to intend that at the time of putting them on
It is not essential in order for it to be permissible to wipe over the socks that one should have the intention. If a person puts on socks when he is in a state of purity, then he may wipe over them for one day and one night if he is not travelling and for three days and nights if he is travelling. For information on the conditions of wiping over the leather slippers and socks, please see the answer to question number 9640.
Shaykh Ibn ‘Uthaymeen was asked: Is it essential, in order for it to be permissible to wipe over the leather slippers, that one should have the intention of wiping over them and also the intention of the period (during which one will wipe over them)?
He replied: In this case the intention is not required, because this is a matter in which the ruling has to do with having socks on, and it does not require an intention. Similarly, if a person puts on a garment, it is not essential to have the intention of covering the ‘awrah during prayer, for example. And when putting on the leather slippers, he does not have to intend to wipe over them or to intend a period for which he will do that. Rather if he is travelling, he may wipe over them for three days and nights whether he intends that or not, and if he is not travelling, he may wipe over them for one day and one night, whether he intends that or not. End quote.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 11/117
And Allah knows best.