Following Thursday's draw, Australian No.1 Nick Kyrgios plays Aleksandr Nedovyesov in the second opening singles on outdoor grass courts.
Sam Groth and veteran Lleyton Hewitt are scheduled to play doubles for Australia on Saturday against Nedovyesov and Andrey Gulubev.
On Sunday in reverse singles, Krygios is set to play Kukushkin and Kokkinakis would play Nedovyesov.
The Australian team has been affected by recent infighting between players and officials. Kyrgios was criticized for his on-court demeanor at Wimbledon, where he was fined for swearing and racket abuse, and Bernard Tomic was suspended from the team for making critical comments about Tennis Australia and its performance director, former Grand Slam winner Pat Rafter.
The countries have never met in the Davis Cup. Players from Kazakhstan competed for the Soviet Union before 1995.
If 28-time champion Australia wins, it will host France or travel to Britain for September's semifinals.
Britain is playing France on grass at London's Queen's Club, the 21st meeting between the countries but the first since 1992.
In the other quarterfinals, Argentina hosts Serbia on indoor clay at Buenos Aires while Canada travels to Belgium to play on outdoor clay. The winners of those ties will play each other in the semifinals, with venues to be determined.
Australia is bidding to reach the semifinals for the first time since 2006.
Kazakhstan has reached the quarterfinals - but never beyond - four of the past five years despite having only one player inside the top 100: Kukushkin at No. 63. Nedovyesov (115th) and Golubev (155th) are the other main team members.
''That is extraordinary,'' Australia captain Wally Masur said of Kazakhstan's record. ''It just points to the focus they have, and intensity and passion.''
Masur said the 20-year-old Kyrgios, who has won both his Davis Cup matches on grass, sustained a right thigh strain in training this week, but has recovered.
''He's fine,'' Masur said. ''He played over three hours of tennis yesterday. He played a bit of basketball this morning. He's got an opportunity to get on that match court tomorrow. No dramas there.''
None of Kazakhstan's team has played a Davis Cup match on grass.
Australia beat the Czech Republic and Kazakhstan defeated Italy in the first round, both by 3-2 scores.
Kokkinakis came from two sets down to beat No.46-ranked Lukas Rosol in Australia's win over the Czechs, and he said Thursday that experience has given him confidence ahead of Friday's opener in Darwin.
''Having that experience in Czech was really good for me - showing I can do it even when things aren't going my way,'' Kokkinakis said.