Firstly, you can't have a two-side arrow in DAG (Direct Acyclic Graph), since it creates a cycle of length 2. Also, I'm not sure if "Error" nodes U1 and U2 are necessary: possible errors can be expressed as noise in CPD (in node "W" and "Y" CPD, to be specific). I will assume next, that U1 and U2 nodes were removed, so we don't have the two-side arrow now.

As for d-separability: you have a direct edge from S to Y; there is no way they can be d-separated.

In general case, the separation set Z is not unique: consider graph A -> B -> C -> D, first set is {A}, second is {D}, and there are multiple sets by which they can be separated: {B}, {C}, {B, C} If you want to find one of them, you can enumerate all the undirected paths from first set of nodes X to the second set of nodes Y, and then separate them, until no non-separated paths are left.

To find out whether two sets of nodes X and Y are d-separated by a set of nodes Z, you can either use the definition of d-separability (i.e. check all paths from X to Y), or the Bayes Ball algorithm.

Note on terminology: this object is called a Directed Graphical Model (DGM) or a Bayesian network, or a belief network, or a causal network. Sure, DGMs are DAGs, but not all DAGs are DGMs; DAG is a far more general term.

P. S. Why Lesswrong, and not, say, StackExchange?