To the two bottom questions: no, and probably not. Jerry Jones' team is a member of a larger association of private organizations that have collectively bargained in order to be exempt from the Sherman Anti-Trust Act. For Zeke, as a celebrity or public figure, he is unlikely to prevail in a cause of action for defamation, such as slander or libel. Public figures, due to their status or celebrity, cannot win defamation suits without proving that the statements were false, known to be false by the person stating them, that the statements were intentionally malicious, and that there was some economic harm suffered. We have the economic harm, potentially, but he won't be able to prove that the statements were malicious, even if they were known to be false. It's very, very hard for celebs to win slander/libel cases.
As far as Zeke's/Cowboys' standing to challenge his suspension, I think it'll be interesting to see what happens. You have a case in which the NFL is essentially claiming that they had more evidence than law enforcement, who declined to prosecute the case due to the inconsistency of statements made by the accuser and other key witnesses. Whether you call it lying or not, without the credibility of the witness at the onset, you're going to have a hard time finding any district attorney who's going to waste the resources to bring charges in a surefire loser of a case.
Goodell has also been wildly inconsistent in handling not only DV cases, but all discipline whatsoever. I think - regardless of whether Zeke's suspension stands - that this term is Goodell's swan song. While I'm sure the owners don't care that much due to profits, Goodell/The NFL has lost 6 of 7 lawsuits brought by the NFLPA for this kind of stuff since 2010's CBA was negotiated. All those L's that ol' Roger has been holding add up to tremendous leverage for the NFLPA come 2020's negotiation, especially in light of recent CTE research and cannabis research. Goodell is officially bad for business for the owners -- even though he eventually won the Brady case, he's costing the owners money with each bungled case. They'd be wise to get him out of there and start fresh with someone else who will put the NFL and the product on the field back at the forefront of the minds of media and fans instead of himself and his disciplinary powers, which isn't fun for anybody except himself.