Why ask here?  This is a board dealing with escorts, not real estate ventures.

You need to contact a lawyer, preferably one that deals in probate/inheritance/real estate law.  Perhaps you may even need to speak with an investments advisor.  Regardless, your post is out of place here.

Your question is not entirely clear.  I am assuming your father has not yet passed away.  If this is the case, and title to the (I am assuming real) property is held by the trustee of a revocable trust, the trustee must distribute that property pursuant to the terms of the trust after a triggering event, such as the death of your father.  The person or persons who receive the property is/are the beneficiary(ies).  Under current law, assuming the triggering event is death of the settlor/trustor (the person who created and funded the trust), the beneficiary(ies) receive a full step-up in basis for capital gains purposes, therefore, there is an advantage to receiving the property on death.  If your father adds you to the deed during life, he is making a present gift that should be reported to the IRS on form 709, and, depending on the value of the property, he may owe gift tax, although the value of the property would have to be substantial.  The more likely scenario is that he would be using some of his lifetime exclusion amount, meaning he could pass less on death without having to pay estate taxes.  Nonetheless, this may be a good strategy if the value of the property is increasing greatly, and is expected to continue to increase because your father would be getting one-half of the future appreciation out of his estate.  Your father needs to discuss this with an estate planning attorney.  As for whether the mortgage company has to offer an assumed mortgage to the beneficiary, the note probably contains a "due on sale" clause, meaning the note must be paid off in full if the property is transferred.  However, due on sale clauses are not enforceable if the transfer is because of divorce or death.  In short, you should be able to assume the current mortgage.  If you have bad credit, the mortgage lender may not forgive even the slightest default, so you must make payments exactly as proscribed under the terms of the note after you inherit the property.  By the way, if your father adds you to title now and you have lousy credit, that could trigger a "due on sale" clause and your father may be forced to refinance to pay off the original note.  An attorney should review the terms of the note for your father.

Laws vary state to state and sometimes even county to county.