In what ways can I factor conditional densities of multiple random variables? For example, in what ways can I factor a conditional density of the type $f(x,y|p,q)$?
If $x$ and $y$ are independent, is it true that $f(x,y|p,q)=f(x|p,q)f(y|p,q)$?
How about $f(x,y|p,q)=f(x|y,p,q)f(y)$, is this true?

In general you can factor a joint density as
$$
f(x, y) = f(x \mid y)f(y)
$$
In a similar way, therefore, 
$$
f(x, y \mid p) = f(x \mid y, p)f(y\mid p)
$$
And
$$
f(x, y \mid p, q) = f(x \mid y, p, q)f(y \mid p, q)
$$
So to critique the examples you gave, you don't have
$$
f(x, y \mid p, q) = f(x \mid  p, q)f(y \mid p, q)
$$
unless $x$ and $y$ are conditionally independent – i.e. it doesn't suffice that they are independent. You also don't have
$$
f(x, y \mid p, q) = f(x \mid y, p, q)f(y)
$$
unless $f(y \mid p, q) = f(y)$, i.e. $y$ is independent of $p, q$.
