Check with an expert--a CPA who specializes in such things. There are a whole bunch of different issues involved here. Taxes, the taxable base value of the property, and more. It may very well be that the best arrangement is not for your dad to give you the house. Possibly (and I don't know; you need a tax/investment professional) some sort of trust. There are trusts that would permit him to live in the unit for the rest of his life, but are far better, from a tax perspective, than his giving it to you today.
And you similarly need advice on the mortgage question. If he were to transfer even part of his ownership interest to you, he might be in violation of the lender's due on sale clause. Depends on when the property was purchased and how the DOS clause was written. Whether the lender would exercise the due on sale clause is another matter. And many real estate investors purchase property "Subject To" (Sub 2) the existing mortgage. So, no, the mortgage would not need to be transferred, even though ownership could be. However, that definitely would trigger any due on sale clause.
However, again, returning to the concept of a trust, the Garn St. Germain act permits an owner to transfer a property into his trust without triggering the due on sale clause.
You've got a number of issues going on here; you need to pay someone some money to give you the best advice.