Properties of a certain matrix in a certain field

In the product $Sx$, $xinmathbf{F}_2^{lvert Ervert}$, the vector $x$ corresponds to a set of edges of $G$. As you've observed, $Sx=0$ means that every vertex of $G$ is incident on an even number of edges of the edge set represented by $x$. The set of edges of a cycle of $G$ has this property. More generally, the set of edges of tour of $G$ has this property. (In a tour, vertices may be visited multiple times, but edges are used only once.) More generally still, the set of edges of a disjoint union of tours has this property. Such disjoint unions are equivalent to edge sets of even-degree subgraphs, or, in other words, subgraphs in which every connected component has an Euler tour.

With regard to the row space of $S$: in the product $x^TS$, $xinmathbf{F}_2^{lvert Vrvert}$, the vector $x$ represents a set of vertices of $G$. The elements of the vector $x^TS$ correspond to edges. As you have found, an element of $x^TS$ equals $0$ when either both vertices on the corresponding edge are in the vertex set corresponding to $x$ or neither vertex is in that set; an element of $x^TS$ equals $1$ when exactly one of the vertices on the corresponding edge is in the vertex set corresponding to $x$. Hence the vector $x^TS$ represents the set of edges that join a vertex in the set corresponding to $x$ to a vertex not in that set. There is an induced subgraph corresponding to $x$, and $x^TS$ can be thought of as the set of external edges—edges connecting to vertices outside the induced subgraph.

The correspondence between row-space vectors and sets of external edges of induced subgraphs is not one-to-one for the following reasons: (1) if $X$ is the sets of vertices corresponding to $x$, the the subgraph induced by $X$ has the same external edge set as the subgraph induced by $Vsetminus X$; (2) if $G$ is not connected then any connected component or union of connected components has no external edges, just as the empty subgraph does. In other words, if $y_1$, $y_2$, ..., $y_m$ are vectors corresponding to vertex sets of connected components, then $y_j^TS=0$ for all $jin{1,2,ldots,m}$ and $$ (x^T+y_1^T+ldots+y_m^T)S=x^TS $$ for any $xinmathbf{F}_2^{lvert Vrvert}$.

So there is a one-to-one correspondence between vectors in the row space of $S$ and equivalence classes of partitions of the vertex set of $G$ into two parts, where the partition ${X,Y}$ is the same partition as ${Y,X}$. The notion of equivalence being used here is the following: let $V$ be partitioned as ${X,Y,C}$ where $C$ is the set of vertices of a connected component of $G$. Then ${Xcup C,Y}sim{X,Ycup C}$.