Rosenstein is career DOJ, not an appointee.
From Wikipedia:
Prior to his current appointment, he served as a United States Attorney for the District of Maryland. Rosenstein was a former nominee to the United States Court of Appeals for the Fourth Circuit. At the time of his confirmation as Deputy Attorney General in April 2017, he was the nation's longest-serving U.S. attorney.
It is correct to say Rosenstein was in the DOJ system at a the time he was appointed. Typically, Federal Prosecutors are required to resign by incoming administrations, but Rosenstein was not required to. He was appointed by Trump on January 13, before inauguration, indicating his hiring as Assistant AG was important enough to be decided early. Trump did indeed choose Rosenstein for the position whereby Rosenstein then appointed Mueller - that is without a doubt.
I have no doubt Trump would like to have that choice back. Trump would probably like to have the Sessions choice back as well. Is Rosenstein a partisan? Is he a mole? Or is he just spineless and won't control Mueller? Nobody knows.
I stand corrected! Rosenstein is career DOJ, but he was US Attorney for MD before his current gig, and was confirmed in April 2017. He must have reauthorized the Page FISA warrant within mere days of assuming his post. Rosenstein is a swamp dweller, but given the circumstance it's possible he may have been manipulated into doing so.
Not a contradiction. DD is an appointment that must be approved by the Senate, but the DD must be from Civil Service. Moves from General Services to Senior Executive Service salaries.