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I'm not sure about the computation piece, but yes, the timeout is supposed to be independent of whether a worksheet is open in a browser or not. In fact, that is sort of the point - if you aren't interacting with the sheet, Sage will close it with that timeout. And if you close the worksheet in the window, you may intend for the computation to continue, so that shouldn't stop the computation.

But this is only a heuristic answer, and others may have more technical replies, in particular about your third point.