Thanks a lot - very interesting. Sounds like this means that all G invariant metrics on a finitely generated group G are in the same quasi-isomorphism class as the word length metrics.

Thanks, Misha, but I'm not sure I see it. Let's assume that $S$ includes a set $S'$ of generators of $H$. It is not in general true that the $S'$ distances on $H$ are the same as the $S$ distances. Intuitively, you could make big jumps outside $H$ using $S$, and then return to $S$ at a further distance than would have been possible using just the $S'$ generators.

Yes, that is true.

It is true that in some sense "conditioning increases divergence", but it is not true (as I note in the question) that the opposite inequality is true. For example, if you condition $X$ on $X$ then the l.h.s. is zero, while the r.h.s. is generally not.