Can a non-Noetherian commutative ring have a nonzero flat nilpotent ideal?
Any such ideal would have to be "tensor-nilpotent" as defined in this question for the same exponent that it is nilpotent. Also, no nonzero flat nilpotent ideal can exist in any Noetherian commutative ring, because any tensor power of a nonzero finitely generated module (e.g. a nonzero finitely generated ideal) must itself be nonzero. In particular, $p\mathbf{Z}/p^n\mathbf{Z}$, where $p$ is a prime number and $n \ge 2$, is not a flat $\mathbf{Z}/p^n\mathbf{Z}$-module.