The 2019 second-rounder will become a first if the Stars win two playoff rounds and Zuccarello plays in at least 50 percent of the team's cumulative games during those series, and the 2020 third-rounder will become a first if Dallas re-signs the nine-year veteran.
The Rangers have also agreed to retain 30 percent of the forward's salary, according to The Athletic's Pierre LeBrun. Zuccarello is in the final year of his current deal with an annual cap hit of $4.5 million and is scheduled to become a UFA on July 1.
The 31-year-old's speed and playmaking ability will provide an offensive boost for a Dallas team that currently ranks 29th in league scoring with 2.51 goals per game.
In the 2014 postseason, Zuccarello helped the Rangers reach the Stanley Cup Final, chipping in 13 points over 25 games. The Norwegian winger has recorded 113 goals and 352 points in 509 career games, all for the Rangers.
The picks headed Ottawa's way are for 2020 and 2021, while Columbus' pick is for this year.
It's the second trade between the two clubs in as many days, with the Senators sending center Matt Duchene to the Blue Jackets on Friday for a first-round pick and a pair of prospects.
Dzingel had been held out of the Senators' last two games as a healthy scratch. As an unrestricted free agent this summer, the 26-year-old was one of the top names on the rental market, and his move to Columbus does not include a contract extension, according to TSN's Pierre LeBrun.
Dzingel is in the midst of a career season and has already set a new personal best with 44 points in 57 games. His cap hit for the rest of the year stands at $1.8 million, per Cap Friendly.
The Blue Jackets jumped into third spot in the Metropolitan Division with back-to-back wins over the weekend to put them at 73 points.
"I kind of expected to be suspended (after talking to the Department of Player Safety, but) I have to be honest, I didn't think (I) was going to be suspended at all going into the hearing," McDavid told reporters Saturday. "But after hearing their tone of voice and whatnot, I had a little bit of a sense that it was going to go that way."
McDavid appeared to criticize the process, which resulted in the first suspension of his NHL career.
"I think a lot of times, they go in and they already have their mind made up," he said. "They don't really care what you have to say. So ultimately, the hearing went as good as you'd expect."
McDavid hasn't decided whether to appeal but has 48 hours from the time of the decision to do so.
Oilers CEO Bob Nicholson said Saturday that the organization is "really disappointed" with the ruling, echoing his statement from Friday in which he said the team strongly disagreed with McDavid's suspension.
Ottawa reportedly made Dzingel and fellow pending unrestricted free agent Mark Stone available for trade Thursday. Both players have been withheld from the Senators' lineup in each of the past two games as the Feb. 25 trade deadline approaches.
Following the Friday trade that sent Matt Duchene to the Blue Jackets, Dzingel now ranks second among Senators players with 22 goals this season.
The 26-year-old winger's expiring contract carries a cap hit of $1.8 million, according to CapFriendly.