For problem 1: I would file a 1099-int from your dad's SSN to the Trust TIN as nominee. This shows the SSN is not receiving and using the interest earned but the Trust TIN is.
For problem 2: You could file another 1041 but if you have already filed your final return, then I would do a nominee from the estate to the trust TIN.
If you are going to do one nominee, may as well do two and keep things simple. To better understand and prepare the 1099-int form, see Topic no. 403, Interest received | Internal Revenue Service
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