Differential Pressure sensing circuit
A very simple reason might be that the differential pressure sensor might be wired backwards and this will cause the AD622 problems on the output without a negative supply: -



The output of the AD622 will swing somewhere between the neg and positive supplies but it won't get very close to them. See the box I've marked in red - it's saying that if your neg power supply is 0V, don't expect the output to get any lower than +1.1 volts in normal circumstances.

The smallest magnitude negative supply ought to be about -1.8 volts to be sure of being able to reach 0V cleanly on the output.

I suspect for your 2nd question that this is all about what happens when the pressure is too big and ch1 on the ADC cannot read it any more - at least ch2 can be relied upon for some readings even if they are subject to a few small extra errors due to resistor tolerances.

For question 3 - I'd like to see a circuit diagram.