1 - Not it isn't. Begs the question is a logical fallacy in which the conclusion is assumed to be correct as part of the reasoning. That clearly does not apply to
So it begs the question of the strength of the NFLPA's arguments before a less biased judge with less history of siding with the NFLPA.
Its a common mistake for people who are attempting to sound smarter than they are.
2- Again that's not what bias means. "Position" or even "tendency" might be a better description except Doty is considered pro-business overall. The NFL has long litigated in the 8th Circuit because its business friendly. The 2nd Circuit, especially in matters of reversing arbitration, is less so.
3 - Your fundamental lack of understanding of this makes me question your legal expertise. The issue is not a matter of precedence in interpretation of a law. Its whether an issue has been settled. Whether an issue has been settled in the 2nd Circuit, the 8th Circuit or a state court, both claim and issue preclusion applies.
Otherwise, Party A could sue in New York, lose on the merits and then simply refile in Minnesota. Obviously this is not how it works. In the 8th Circuit, Judge Doty determined that the NFL is prohibited from suspending players for behavior that is not detailed in the CBA or in a manner that contradicts the prescribed punishments, ie that a player must be given notice that his actions may result in punishment and the kind of punishment likely to occur under Article 46.
The NFL's case argues that the Commissioner is not required to give notice and this matter is central to Brady's appeal. This issue has been decided in federal court. The NFL is thus prohibited from arguing that they don't have to give such notice. Without this point, their case has no coherence.
What you've done is basically this. Person A and Person B get into a car accident. Person A says "Person B blew the stop sign." Person B says "Person A blew the stop sign." They sue each other. You say that Person A is right. Why? Because Person B blew the stop sign.
No its more like Person A was found by a federal judge to have blown through the stop sign and is now attempting to argue in a subsequent case that he didn't. That's not how it works.