Cycle cover for structured graph
Re your first question, re whether this type of graph has a name: it is the line graph of a bipartite graph (the bipartite graph that has a vertex for each nonempty row or column and an edge for each point). Line graphs of bipartite graphs are an important subclass of the perfect graphs. They are generally not planar (e.g. if you have five points on the same row or column you get a $K_5$ subgraph).