Ok so in my own defence... Giovanni is the highestranked in squad 5. I operated under the assumption that only captains and lieutenants could give missions so I didn't havgo handle that. 

Also in regards to his work. Prior to the year time skip he bothered the head captain and all that.. since the time skip however he has had a much more independent role. In speaking with kurisa it was established that the head captain lead the kido corps. And then that it wold operate under squad five. 

In squad five it was his responsibility to make sure the two units blended well and created cohesion. Which he did.he handles most of the paper work for squad five responsibilities as well as maintains authority in the kido corps. 

You are right Abi, he does not lead the kido corp, because we have no record of his rank there. But kurisa and I spoke around the time of the year skip and established that the kido corps coming out of complete secrecy and blending g with squad 5 was due in large part to giovannis efforts and work. 

As for going to squad thirteen for the recommendation there is no captain or lieutenant for squad five. According to the OP rules for promotion go only needed a recommendation from A captain. Not necessarily their own captain. While in Canon it is required for it to be the captain of the squad...there is no captain of squad five.