I'm not keen on "through from under"; it sounds like the person doing this is actually under the door themselves. I think you mean "through under", but "under" by itself will probably suffice. Everything else is fine.
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Mr. TomSo, are these perfectly OK? If I understand you correctly, there is no need for "through" at all?Yes, these are fine. It is also OK to say "through under" in both cases. "through" doesn't change the meaning a great deal; it just give a bit more
He slid the newspaper under the door.
Please don't ring the bell. Just slide the bill under the door.