  We give examples of rank-one transformations that are (weak) doubly ergodic
and rigid (so all their cartesian products are conservative), but with
non-ergodic $2$-fold cartesian product. We give conditions for rank-one
infinite measure-preserving transformations to be (weak) doubly ergodic and for
their $k$-fold cartesian product to be conservative. We also show that a (weak)
doubly ergodic nonsingular group action is ergodic with isometric coefficients,
and that the latter strictly implies W measurable sensitivity.
