so this is my own deduction.
I don't think yeon was reincarnated. Instead he was summoned back by (the last judge) king of darkness who in-charged of reincarnation, but it come with a price. the sword is for killing imoogi only. when yeon and imoogi fell into samdo river, only imoogi were disintegrated so he will never reincarnate, but not to yeon. so apparently yeon was never died from the start.
the judge did say that people get reincarnated randomly. so this, might be that reincarnation could happen in any time with different or same face, different gender, things and even into animal. this could be the answer to rang's incarnation. remember soo ho was a puppy back then.
when yeon know that he would have a daughter, that might be the sign that he was never human actually. that time he might thought that was just an intuition. but i think he might be very sure that he turn into human at first soon after he being pulled from samdo river because he thought he had no power anymore that he even got astigmatism and toothache. samdo river might have naturalized his power for the moment and somehow he got it back as time passes.
granny back to normal might be after imoogi died in the river since it's imoogi who cast her into stone. also don't think that yeomra will let her sister died for nothing.
bok gil also might be disintegrated when he killed himself (suicide). bok gil's story itself already mystery since imoogi also being involved in it. so let just be it. it was another story anyway.
yeon was never half human/fox by the way. he already lost his mountain god status back 600 years ago. if he still mountain god, he wouldn't have go to do the cleansing evil spirit instead.
actually when yeon stood in front of his umbrella he kind of hesitate. somehow he looks like 'ah nevermind, let's just try it. this might be still useful. or else, why the umbrella is still back to me.' the umbrella is still with him might be to shows that he is still a deity/fox.