The seventh-seeded Spaniard was playing her first match since losing in the second round of her Wimbledon title defence.
Around 120 delegates at the International Tennis Federation's annual meeting in Orlando will decide whether proposals by federation chief David Haggerty are to be given the go-ahead.
Federer's defeat of the 47th-ranked German in 72 minutes marked his first step towards a possible eighth title at the tournament.
Wawrinka, who beat Novak Djokovic to win the title in New York in 2016, has dropped to 151st in the world after a knee injury that ruled him out of the second half of last year.
But the 36-year-old was all business Monday as she cleaned up on Australian Gavrilova, with Williams winning an 11th straight match here.
Murray, returning from January hip surgery, was unable to get past the French 16th seed, hampering his preparations for the US Open starting on August 27.
The French Open champion beat world number three Sloane Stephens 7-6 (8/6), 3-6, 6-4 in a gruelling 2hr 41min battle.
The world number one Spaniard denied his Greek opponent a major upset on the youngster's 20th birthday.
World number one Nadal won a battle of the tennis generations with Tsitsipas last April in the Barcelona title match.
The proposal must win approval at the federation's annual general meeting in Orlando, Florida this month.
The top-seeded Halep hammered 19 winners and won 77 percent of points on her first serve in the 88-minute contest.
The Greek teenager, who turns 20 on Sunday, defeated his third straight top 10 opponent after knocking out Dominic Thiem.
Anderson hadn't defeated the Bulgarian for six years, but he was in command throughout on Friday.
When the weather passed, the pair returned, with Nadal wrapping up the set on his second chance.
Top seed Halep took no prisoners against Williams, blitzing the 38-year-old American 6-2, 6-2 in just 71 minutes to reach the quarter-finals.