What? That post clearly says that Hunter isn't buying the house, he's making the mortgage payments.
That means that no, Hunter does not own the house. He couldn't even legally buy the house without Ophelia's and the bank's permission anyways as they are under contract with the mortgage.
Also, if Hunter owned the house, that would mean that he would not need to make any sort of mortgage payments.
But he doesn't have to pay Ophelia directly either. You can make payments on other people's behalf. Though, if there's no sort of contract, which I doubt there is, he actually wouldn't have any real legal claim to the house either.
But he also wouldn't want to put his name on the mortgage either. If he did, then the bank could legally come after his assets if he elected not to pay.