Hogs'-backs are hills.
The Hog's-back is a raised part of the North Downs in Surrey. Hardy is using the term in a more general sense here to describe any hills, ridges or escarpments that mimic the narrow, rounded shape of a pig's back.
Alec is referring to his former self, as he was before he converted to Christianity.
The term "old Adam" was a common short-hand for the evil that is supposed to be inherent in human nature and can only be washed clean through baptism.
A quotation from St Paul's second letter to the Corinthians (6:17) in the New Testament. Theologians differ as to how to interpret these words. While some believe they advocate closed religious communities, others read them as injunctions to keep separate mentally and emotionally from the ways of the secular world.