Can you explain likelihood function in the equation?
To me, it looks like $\theta$ is just a throw-away "any variable". They might just as well have written $r(s|\cdot)$.

Equation 12 is just saying that because $r$ and $s$ are mutually independent, the distribution of the error $\rm\bf e$ does not change. Compare (12) with (10) rearranged to give an expression for $\rm\bf e$.