Federal statute can. The "Byrd Rule" is Federal law(2 U.S.C. § 644) in the form of an amendment to the 1974 Congressional Budget Act of 1974 which was passed by Congress in 1990. It's not just a Senate rule that can be modified at a whim.
Nope.
In order for the Byrd rule to bind the current congress, it would have to be a constitutional amendment, not a mere federal statute, much less a Senate rule (Senate rules can be changed by a majority, even Senate rules requiring a supermajority — thank you, Dingy Harry!).
Congress makes the law by majority, with the approval of the President, unless two thirds of both houses vote to override his veto. New law supersedes old law (provided it's constitutional).
Thus, if the Senate were to pass a law in violation of the Byrd Rule, and the House passed it, and the President signed it, it is the law, Exalted Democrat Cyclops Byrd be damned!