Algebraic group actions and Zariski's cancellation problem

I call an algebraic variety flattenable if it is isomorphic to an open subset of an affine space. I shall discuss, in the frame of algebraic group actions, several topics which stem from the following question resembling Zariski’s cancellation problem: Are there affine algebraic varieties $X$ and $Y$ such that $X$ and $X \times Y$ are flattenable, but Y is not? Among them is the problem of classifying a class of affine algebraic groups naturally singled out in the study of algebraic subgroups of the Cremona groups.