In terms of mimkomcha, the story I heard, possibly from SC himself or from one of his students (It was a long time ago and the story is fuzzy so don’t take it as 100%) was when SC sang it for the Amshnav Rebbe zt”l (I think) the Rebbe said that this tune is nusach from the beis Hamikdash.
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The Amshinover Rebbe may hve liked the piece. In fact I do too. It just has no place being sung during davening. As a folk tune, it’s great though. I can prove that such a meoldy was NEVER heard in the Bais Mikdash. I know this becuse there is absolutely no similarity to trope in the melody.