Re: follow somebody's cue = take the cue from somebody
Your interpretation looks fine to me.
As you probably already know, a "cue" (either as a stage direction, or as a particular word or phrase in the script) marks the point at which an actor should say or do a particular thing.
Thus some word or deed of Putin's indicates to Medvedev that he should empower the FSB; and some coarseness of Ben's, according to Sally, leads her into something similar.
Not a professional ESL teacher.