The DoF app I have calls for the sensor width to be set. The specs for my Sony α33 say the image format is 23.4 x 15.6mm. Do I dial-in the 23.4 or the closest to it? If I must go slightly over/under, which should I use?
Either value that close to the actual measurement will be fine. Much more important is making sure the app also knows the correct display size and viewing distance you intend to use. Those factors, which are often assumed to be a specific value and any deviation from those assumptions is ignored, are much more frequently the cause of invalid results when calculating Depth of Field.
Your question appears to reflect the common misconception that everything within the Depth of Field (DoF) is equally in focus. This is not the case at all. There is only one plane that is in focus for any position of your lens' focus mechanism.
In a way, depth-of-field is an illusion. Everything in front of or behind the point of focus is out of focus to one degree or another. What we call DoF is the area where things look, to our eyes, like they are in focus. This is based on the ability of the human eye to resolve certain minute differences at a particular distance. If the slightly out-of-focus blur is smaller than our eye's capability to resolve the detail then it appears to be in focus. When you magnify a portion of an image by making it larger or moving closer to it you allow your eye to see details that before were too close together to be seen by your eyes as separate pieces of the image.
Since things are gradually blurrier the further they are from the point of focus, as you gradually magnify the image the perceived depth of field gets narrower as the near and far points where your eyes can resolve fine details moves closer to the focal plane. That is why DoF calculations must include the display size, viewing distance, and acuity of the viewer's vision in order to give a meaningful result.