To answer your last question in the simplest way, yes you can. But make sure your understanding is clear with your father and wife about what the payments will cover and ultimately what will happen to it upon satisfaction of the mortgage or his death, etc. For example, what would stop him from conveying it to his new girlfriend half his age who sweeps him off his feet next year? Even if he promises the house to you what assurance do you have that it will happen that way (not much)? You could have him convey a life estate to you but that creates a problem between him and the mortgage company.
As to whether this is the most cost effective/efficient it's hard to say. He won't get the mortgage interest deduction if you're paying it, but that could be minimal and irrelevant if he doesn't itemize.
This is only the tip of a few thoughts on what I see as an iceberg - not necessarily a dangerous one - but something that should be carefully thought through. My advice would be to consult with a tax professional/attorney.
Let me break you question down into smaller pieces.
First of all, if you are selling you own home, make sure you lived in the home for 2 of the last 5 years so that you get the IRS break on the first $500k of gain.
Next you need to understand that your father will be making a gift to you of his equity. You will have to file a federal gift tax return. There should be not tax, but the return is needed. Also, you need to be careful about the look back if your father goes into a nursing home and is covered by Medicaid.
Thirdly, you will need a deed. This will trigger a new homeowners policy and notice that your name is on the policy to the lender. They will send you a letter pointing out that paragraph 18 of the mortgage is a "due on sale" provision and that you need to pay them off. You will reply that your father is still there and that you have moved in to care for him. Depending on the lender, you may find yourself learning to do the Tango or some other dance with a servicing agent in the Philippines or in India.
Good luck. As you can see, you are going to need a lawyer to do the documents. You might as well get some planning advice at the same time
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As the other attorneys have said, the most cost efficient may not be the most effective. These are very tangled issues and must be looked at from all angles to avoid some very unpleasant unexpected consequences. Make sure the attorney you choose has real estate experience, not just estate planning. What will be most important is to have all of the transaction documented so that there are no surprises later.