Oznog, I'm also baffled. Your basic question has nothing to do with the rest of your components. As I understand it, you simply connected the input pin to the GND pin, at the IC,, and got an output code which indicated 1.3mv on the input. This is, as you said, way outside the spec, which is 2ppm of Vref. You implied that Vref is 5v, which should yield a maximum offset of 10uV. My calculations show that the maximum code with zero input should therefore be +/- 34 LSB's, or 22H. Your reading of 1.3mv would be 110AH. Am I interpreting the spec and your reading correctly?
Regarding the precision of your other components, It sounds like Vref and the current sense resistor are all you have that figure into your calculations. Both of them contribute to scaling errors, but not to offset errors. When you are trying to measure from milliamps to 100's of amps, I can see why offset would be your primary concern.