I am Beneficiary, Successor, to property left. What can I do to seal title?

Your questions cover many areas of Probate and real estate law, and you can’t get good answers until you consult with an attorney in your state who understands all the circumstances.

You say that you were given the house. How? In a Will? Trust? Life Estate Deed conveyance? You also mention that there is a 2nd mortgage and back taxes. Are you setting yourself up for financial problems by accepting ownership of the property? A careful look at the encumbrances (debts) on the real estate is required, before you make a decision about what to do next.

If the property is encumbered by substantial debt, it may not be worth it for other people to contest your ownership, if you do have claim for ownership. But you won’t know where you stand until you get professional legal advice.

What does the will (or trust) provide? That's (they are) the governing document(s), unless your friend deeded the house to you prior to passing.

Also, you wrote you're "successor" - successor trustee? What kind of "successor" are you? I'm not really sure what you mean.

You wait for the probate to be completed. Even a small estate with no Will has to have a fiduciary or executor. The person who executes the Will must first wait for all claims against the estate. In NY the waiting period is 7 months. The executor must pay all bills presented, including that mortgage and taxes. The bank holding the mortgage will either foreclose or agree to a "short sale".
You say he gave you the house, but you don't say HOW he gave you the house.
A written promise? A quitclaim? An addendum to his Will? He just let you move in with no paperwork? You may need a lawyer to sort this out.

If by "seal" you mean getting it notarized, it is too late for that. Sorry.