Smith, 35, was 19-26-9 with a 2.92 goals-against average and .914 save percentage in 55 starts with the Coyotes this past season. He has two years remaining on a six-year, $34 million contract.
Smith totaled a 128-132-41 record with a 2.69 GAA and a .916 save percentage in 312 career games with the Coyotes over six seasons.
"On behalf of the entire organization, I'd like to thank Mike for everything he's done for our franchise on and off the ice the past six seasons," Coyotes general manager John Chayka said in a statement. "He's the consummate professional and we wish him all the best in the future."
Johnson, 31, posted an 18-15-1 record with a 2.59 goals-against average and a .910 save percentage with the Flames in 2016-17. He can become an unrestricted free agent on July 1.
The 21-year-old Hickey recorded four goals and 11 assists in 35 games as a junior with Boston University in 2016-17. He totaled 15 goals and 25 assists in 112 career games.
If Calgary qualifies for the Stanley Cup playoffs in 2017, the Coyotes will get the Flames' second-round choice in 2018 or 2019 NHL Draft (at Calgary's option). If Calgary does not qualify, the Coyotes will receive the Flames' third-round choice in 2018.