Without looking it up again -- I protect most of my clients from living trusts, so I don't deal with this question too often -- the trust can have an EIN but no 1041 filing requirement unless the trustee is not the grantor/beneficiary. Back in the old days, it was more common to get an EIN even if not needed.
You cannot put an IRA into a trust. An IRA is already a trust. You can name a trust as beneficiary of an IRA, but generally that is foolish and can cost thousands in taxes. If you look at the 1099-R for the IRA distribution, you will see her SSN.