Is it a Rail-To-Rail opamp?
Why opamp doesn't behave as datasheet "guaranteed"?


It does, but only the values in the tables are guaranteed. The rest is, well, up to anyone's interpretation.


  How can I achieve 0V output with this opamp?


Maybe by loading it with only resistor to ground. 9 mV sounds quite good to me already. If you really need to go to 0 V, use a negative supply rail!


  If I connect opamp ground to -1V it would work?


Yes, but don't call it "opamp ground" but call it "negative supply rail". A ground should have 0 V, you suggest -1 V so then it's no longer a ground.