Answer:

$\dim{K} = \dim(S)$ because $f = \frac{S} {S + K}$
f is dimensionless because $f = \frac{S} {S + K}$
$\dim(j_S) = \dim(S)/ (\mbox{time} \, \dim(X))$
$\dim(y_{XS}) = \dim(X)/\dim(S)$
$\dim(\dot{r}) = \mbox{time}^{-1}$

A possible choice for the dimension of S and X is: $\mbox{C-mol} \, . \, \mbox{length}^{-3}$. This does not imply, however, that y_XS is necessarily dimensionless; we have $\dim(y_{XS}) = \frac{\mbox{\footnotesize C-mol X}} {\mbox{\footnotesize C-mol S}}$and $\dim(j_S) = \frac{\mbox{\footnotesize C-mol S}} {\mbox{\footnotesize time} \, . \,\mbox{\footnotesize C-mol X}}$.