Living trust filing question

I am the successor trustee for a taxpayer with a living trust who died in mid-December after receiving all income for the year. 

(1) Do I file a 1040 or 1041? (A paid tax preparer prepared a 1040, but I'm not secure that this is correct.)

(2) Do I use the taxpayer's SS# or the EIN for the trust, received after taxpayer's death?

(3) If I'm filing a 1040, how do I show the IRS that I am the trustee and have the right to sign the return?

 

Thank you.