Rafael Nadal, Naomi Osaka and Milos Raonic are the high profile names to opt against playing at Wimbledon which starts 28 June.
The 35-year-old will miss clay court events in Marbella, Monte Carlo and Belgrade in April.
World number three Medvedev was the top seed for this week's tournament following his run to the Australian Open final but was dumped out 7-6 (7/4), 6-4 in just over 90 minutes by Serbian Lajovic, currently ranked 27th.
Big-hitter Khachanov eased through the match without dropping a service game to eighth seed Wawrinka.
Novak Djokovic’s Australian Open dominance improved to 18-0 combined in semi-finals and finals. On Sunday, Daniil Medvedev joined the ranks of Roger Federer, Rafael Nadal, Andy Murray, Stan Wawrinka, Dominic Thiem, all of whom have lost to Djoko in semis or finals in Melbourne.
Australian Open 2021: 'Feel like dying' Marton Fucsovics after yet another five setter; Serena Williams reveals who she wants to meet
Fucsovics has now survived two five-set matches in his first two rounds, spending more than eight hours on court.
Australian Open 2021: Djokovic, Serena win; Wawrinka squanders three match points, Andreescu's comeback stalled
Though Tiafoe played an aggressive match, Djokovic was better when it mattered. He had the edge in terms of winners (56-49) and converted five break points to just two for Tiafoe.
Australian Open 2021: Serena Williams, Naomi Osaka sail into second round; Angelique Kerber, Gael Monfils out at first hurdle
Venus Williams, on Monday, won a Grand Slam match for the first time since 2019.
Seeded one at the Murray River Open and in his first match of the season, Wawrinka lost the opening set against his Kazakh opponent but kept his composure to fight back and win 4-6, 6-3, 6-1.
World number 18 Wawrinka explained his physical preparations for the season-opening Slam had been affected by the setback.
Novak Djokovic has already clinched the year-end number one ranking for the sixth time as he heads into the ATP Masters starting at London's O2 Arena on Sunday.
Nadal, who claimed his 1,000th career victory by beating Feliciano Lopez on Wednesday, is bidding for a first Paris indoors triumph which would draw him level with Novak Djokovic on a record 36 Masters titles.
Paris Masters 2020: Rafael Nadal romps to 1000 Tour-level wins; Diego Schwartzman inches closer to ATP Finals debut
The 28-year-old Schwartzman was too strong for France's Gasquet, taking a 7-5, 6-3 victory, and will next face Spanish qualifier Alejandro Davidovich Fokina in the last 16 at Bercy Arena.
Humbert, who is on a seven-match winning streak after winning the Antwerp title last month, missed four match points but battled to a 7-6 (7/4), 6-7 (6/8), 7-6 (7/3) victory.
Top-seeded Djokovic lost four straight games and had to save a set point before beating Filip Krajinovic 7-6 (6), 6-3 in an all-Serbian match, and No. 2 Thiem nearly squandered a 5-1 lead in his 6-4, 7-5 win over Vitaliy Sachko of Ukraine.
French Open 2020 Round-up: Rafael Nadal eases past Mackenzie McDonald; injury forces Serena Williams withdrawal
Azarenka hit 38 unforced errors in a 6-2, 6-2 defeat against a player who hadn't won a Grand Slam match in five years before her first-round victory over Venus Williams.
French Open 2020: Andy Murray says 'difficult' to be former self but vows to 'keep going ' after Stan Wawrinka rout
Andy Murray suffered his joint-worst defeat at a major in terms of games won, succumbing to 2015 Roland Garros winner Stan Wawrinka 6-1, 6-3, 6-2 in the opening day’s marquee match of French Open.
Rafael Nadal, who is seeded second, is aiming to equal the absent Roger Federer's men's record of 20 Grand Slam titles.
Wawrinka, now 35, ranked as high as third in the world in 2014 when he won the Australian Open. He went on to win at Roland Garros the following year and the US Open in 2016.
Italian Open 2020: Teenager Lorenzo Musetti stuns Stan Wawrinka; Garbine Muguruza beats Sloane Stephens in straight sets
Stan Wawrinka, the 10th seed in Rome, lost 6-0, 7-6 (7/2) against a player ranked 249th in the world.