Splitting up A Conditional Probability When, if ever, is it true for random variables $a,b,c,d$ that $P(a,b,c\mid d) = P(a\mid d)P(b\mid d)P(c\mid d)$?  Is it related to the dependencies among the variables?

It can be true, and yes, it is related to the dependencies among the variables. It holds when variables $a$, $b$ and $c$ are conditionally independent given $d$.
