Of course, you're right that what mathematical structures are the equivalent of the everyday terms we're talking about here depends on exactly what we're trying to accomplish. (Though note that the definition of trapezoid versus a square inherently makes use of lengths or angles hence implies we need to be working on geometric and not purely topological structures to be able to make any statement about trapezoids that are meaningful. I.e. a trapezoid can be the domain of a covering of a torus but only when we are interested only in the topology and not the geometry and hence only when "trapezoid" is indistinguishable from "square.")

Yeah coverings can't be isometries (unless they're trivial), but they can always be local isometries which is what we'd most likely want here. R^2/Z^2 is (isometric to) a flat torus, which is, I'd argue, the nicest (though not the tastiest) torus. (Though the question of which flat torus is nicest is another matter and many would say it's not the torus that comes from the square!)

I don't know of any general statement for you about what pulls back by coverings but in general anything you can describe as "local" pulls back. That's pretty vague though, sorry!