In a barometer, how important is it that the inner diameter of the barometer be uniform? That the barometer tube be absolutely vertical?

Pressure depends on the height and density of the liquid used in the barometer. The height of the liquid column is due to the pressure exerted by the atmosphere on the other surface of the barometer. It is not depend on the diameter of the barometer.
But the pressure difference between the tube is directly proportional to fourth power of its diameter. So, the pressure difference between the narrow regions is different from the pressure difference between the wider regions of the tube used in the barometer. In order to avoid such discrepancies, the diameter of the tube should be uniform.
For accurate measurement of the pressure, the tube should be absolutely vertical. This makes to determine the height of the liquid column easily. If the tube is not vertical but inclined to some angle, the corresponding height should be taken the vertical component.