That's adorable, and clearly a construct very much like German (or Yiddish, obviously). My wife's Pennsylvania Dutch grandfather used to say things like that.
And I love the term "Yeshivish." Is it different from Leo Rosten's old "Yinglish," or is that term just outmoded now?
As an American (California), saying "at mine" sounds wrong unless it's in response to a statement that included the relevant noun. i.e. "Is he staying at her place?" could be answered by "no, at mine", but you would never say "he's staying at mine" without that noun being stated first.
Huh? What does they stayed at mine even mean? They stayed at my house, or with me, or at my place, or even by me, I would understand. But they stayed at mine? Unless someone asked, whose house did they stay at and the answer was they stayed at mine. But if the question was where did they stay, you wouldn't answer at mine. You would say my house. Totally weird translation
Well, it would work if we assume that they were already discussing whose place they stayed at or someone had just asked "So whose place did they stay at?"—or, if anyone prefers, "So at whose place did they stay?"). "They stayed at mine." It's taken for granted they stayed at someone's place but "whose"—that's the salient point.