The problem with your premise here is that Arpaio is not technically acting on behalf of the state itself. As a county level official, Arpaio does get his authority as sheriff from Arizona statutes and/or the Arizona Constitution. However, he is technically not a “state” level official nor a state level actor. Therefore, his lawsuit does not technically fit the bill to be classified as original jurisdiction for the SCOTUS.
You are correct, but at this stage of the game who cares about the rules anyway? Which means Governor Brewer should have joined the case first. But she can always file NOW directly with the supreme Court and if they refuse, she is free to do as she wishes, simply because an inferior Court has no authority over a Free State. But I also said Governor Brewer should have told the Judge in the Drivers License for Illegals case , to go pound sand, there as Governor, She is the “State”.