Hello Bruce, welcome to Using English!
I would agree that those sentences are rather strange. As you suggest, the correct forms would be:
1. When he (de)parted that morning from the old man's cell, he paused to
behold him | who held open the heavy-timbered door.
2. "Woe unto him | who rises in the morning to following strong drink."
It seems to me that the writer may indeed have decided that the case of the relative pronoun should somehow affect the case of the antecedent; on which basis he would presumably recast (googling at random on "he whom"):
3. "For he whom God hath sent speaketh the words of God".
4. "For him whom God hath sent speaketh the words of God."
Or perhaps it was a case of hypercorrection, after the pattern of:
5. "He who laughs last laughs loudest."
On your second question, I would take "my", "your", "his", etc. as "possessive determiners" or "possessive adjectives", while "mine", "yours", "his", etc. are possessive pronouns, e.g.
6. It's my book, not yours.
Here, "my" functions as an adjective, but "yours" functions as a noun.
Have a good evening,