you're getting the burden of proof backwards.
This is incorrect. Under US law, laws and regulations subject to rational basis review are presumed rational. The challenger has the burden of proof.
again, you were backwards. you started with 'yes, unless' when the real answer is 'no, except.' that's not a minor distinction.
Again, this is factually wrong.
For details: www.law.cornell.edu/wex/rational_basis_test