Is hair extension permissible?

All Perfect Praise is due to Allah, The Lord of The Worlds 

It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.

Should vowed fast be offered in consecutive days, or not?

Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.

Does wiping over one's socks during ablution permissible?

If the socks were thick,  then wiping over them during ablution is permissible so long as wiping conditions are observed. And Allah Knows Best.

Is it permissible to offer Istikhara prayer (seeking guidance) after Witr Prayer (A confirmed Sunnah which is offered after Isha prayer and before Fajir prayer with a minimum of 3 Rak`as and a maximum of 11)?

It is permissible to offer Istikhara before and after Witr prayer as it is permissible at any time except that in which prayer is disliked. And Allah Knows Best.