Swiatek rallied from a set and a break down in brutal heat to overpower Estonian Kaia Kanepi 4-6, 7-6(2), 6-3.
The 25-year-old reached the semifinals at Melbourne Park for the second time in three years after blitzing American Jessica Pegula 6-2, 6-0 in just over an hour.
The unseeded Indo-American duo went down to the Australian pair of Jaimee Fourlis and Jason Kuble in the quarterfinals.
After dropping the fourth set, Nadal marshalled his resources off court with a medical time-out then returned to break Shapovalov in the second game to wrest back the momentum.
Keys booked her first Grand Slam semifinals in four years with a 6-3, 6-2 upset of Krejcikova.
Swiatek rallied from a set down to edge Romania's Sorana Cirstea 5-7, 6-3, 6-3 in a baseline slugfest.
Medvedev teed off at the chair umpire and lost his cool in the heat before steadying to claim a 6-2, 7-6(4), 6-7(4), 7-5 win over Maxime Cressy.
Frenchwoman Cornet, making her 63rd main draw appearance at a Grand Slam, improved her record against 2018 finalist Halep to 4-1 after the 32-year-old came through a battle lasting two hours and 33 minutes at the Rod Laver Arena.
Evert said a malignant tumor was discovered during a preventive hysterectomy early last month.
"A very small majority chose to follow their own way, which kind of makes the majority look like fools," Tsitsipas said.