I’m guessing that the answer may vary depending on the constitution of any particular state.
Having said that, I’m going to go on a WAG jag.
In most states, the attorney general is a constitutional officer elected directly by the people. As such, the A.G.'s authority is independent of the authority of the governor – that is, the A.G. does not take orders from the governor.
As for No. 2, I don’t know for sure, but I’m guessing that only the attorney general can bring a suit on behalf of the state. The governor could appoint an attorney only to represent himself.
In Colorado, to a large extent, this is true. There was another thread recently that showed that in California, to a large extent, this is not true. It does all depend on the state.
I would be interested in seing that thread about CA, because in the recent flap over SF issuing marriage licenses to same-sex couples the CA AG made it quite clear that he does not take orders from the governor.
In Wisconsin at least, the governor can take legal action on behalf of the state independent of the AG. Our last governor, Tommy Thompson (now HHS secretary) had something of a history of hiring private attorneys to defend state statutes when the AG, Jim Doyle (now governor) declared that his office wouldn’t defend them since they were likely to be found unconstitutional. The final tally was IIRC 7-0 in favor of Doyle.
I’m not convinced based on the thread that the governor of CA can order the AG to prosecute or defend a particular suit. The AG certainly has a duty as the state’s lawyer ( and I think calling the AG “the governor’s lawyer” was just a rhetorical thing in the quoted story) but I don’t see based on the language quoted that the governor can step in and dictate the AG’s actions.
Also not convinced by Truth Seeker’s linked story of, well, anything, but that’s an issue for another thread.