In general, money or assets received after the date of death belong to the heirs, to be distributed according to the decedent's will. If, however, the income or assets were generated by investments or property owned by the Trust, they would belong to the trust and be distributed to the beneficiaries according to the trust documents.
The fate of any money or assets owned outside the trust would be governed by the decedent's will.
And, yes, you would complete the final trust return and include Form 1041 Schedule K-1 in the final Form 1040 tax return for the decedent.
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