Towards the end of GFIII, we see Neri and Mike's old bodyguard address Vincent as Don Corleone. My questions is, is Al Neri now 2nd in command to Vincent? If so, who looks after Mike, even though he's retired? Also, why does Mike yell at Neri, Vincent and Connie after the attempted hit? He makes Neri say "OK" after the "I control this family, right or wrong" speech. What's that about? Did he feel his main caporegime Neri was pulling too much weight?

At the end of GF III, I think Al Neri is second-in-command to Vincent, but I think Michael and Connie are both functioning as consiglieres/ Dons Emeritus to Vincent. Connie even "makes her bones" by killing Don Altobello.

I think it's fair to assume that whatever structure the family has after GF III, Michael and Connie are both going to be part of it.

The earlier scene, after the attempted hit on Mike, is Mike reasserting his control. He doesn't want Vincent, Neri or Connie to think he is old and helpless. At that point I don't think he believes Vincent is ready for that kind of authority.

Excellent answer, Enzo! I'd add only that Vincent assumed responsibility for protecting Michael, and Vincent brought in the "Best" twins to bodyguard him.