Gough, 51, who had two playing spells with Yorkshire, captaining them in the second, has been appointed initially until the end of the 2022 season.
Root, who said he had exchanged messages with Rafiq since he gave his evidence, insisted again on Monday that he had never heard anything untoward.
Following Rafiq's comments, which have sparked a tidal wave of accusations of racism within English cricket, the 47-year-old Vaughan was stood down from his BBC radio show earlier this month.
Cricket NSW chief executive Lee Germon, who oversees Sydney Thunder said the Australian T20 side should "accept people who have made mistakes".
Gayathri Ajith was only 16 when she met Rafiq but said she told him she was 17 in order to appear a "bit older".
The apology comes amid an ongoing crisis in English cricket sparked by whistleblower Azeem Rafiq's revelations about the racism he suffered while playing at Yorkshire.
Brooks, who now plays for rival county Somerset, also apologised Thursday for using the word "negro" in a conversation on Twitter nine years ago with England bowler Tymal Mills and Stewart Laudat, with whom he played minor counties cricket for Oxfordshire.
Yorkshire row: Azeem Rafiq tells British lawmakers racism institutional in cricket; he felt 'humiliated'
The 30-year-old Rafiq said racism had been commonplace at Yorkshire, one of England's oldest and most successful clubs
The 33-year-old Rashid, who was a key member of the England team that reached the semifinals of the just-concluded T20 World Cup, said he will be "happy to support" any further official investigations into Rafiq's claims
John Faragher, who stepped down on Thursday, denies the allegations against him, which relate to an Essex board meeting in 2017, but the club are looking into why the incident was not fully and independently investigated at the time.
Lee said in the interview posted Wednesday that she and friends were waiting for an Uber ride when a car drove by with its occupants shouting slurs and that she was pepper sprayed on her arm as the car sped away.
Former player Azeem Rafiq filed a legal claim against Yorkshire last year, which has now been settled, for failing to deal adequately with racist abuse he suffered while representing the county
English cricket has been plunged into chaos after Pakistan-born off-spinner Azeem Rafiq accused Yorkshire of failing to deal adequately with allegations of racism when he played for the club.
Hutton said that there has been a constant unwillingness from the executive members of the board and senior management of the club to apologize and to accept that there was racism
Proven allegations of racist behavior or the use of racist language could have serious implications for Sarver.
Alongside De Kock, who returned to the playing XI for the Super 12 game against Sri Lanka, his South African teammates and the on-field umpires were also seen taking the knee, in support of the 'Black Lives Matter' (BLM) movement.
Rafiq made serious allegations about his experiences at the club and Yorkshire appointed lawyers to launch an investigation, with an independent panel also put in place to oversee it.
T20 World Cup 2021: Quinton de Kock has put the controversy surrounding taking knee to bed, says David Wiese
De Kock came out with a statement on Thursday saying he is fine taking a knee if it "educates others" and that he was deeply hurt at being called a racist for refusing to make the gesture earlier.