None of these are true in Routh's case. That makes him a non-Native with a small amount of Native blood. Same as tens of millions of other Americans, none of whom qualify as "Native."
We're talking about why the role didn't go to one of the talented Native actors out there who looks and acts Native and represents his Native culture. Why are you defending Routh when he's taking a job from a real Native actor? Should Adam Beach, Chaske Spencer, et al. stand on the sidelines and applaud Routh the "Native"?
Someone else added:
For more on casting decisions, see Tavare on Hollywood Indians and Depp's "Dilemma" Over Playing Mexican. For more on who counts as an Indian, see Indian Identity Matters to Indians and "Actual Indian" Defined.
Below: The next role for Common the "white" actor?