In answer to your first question, you might want to come at it from another angle (if you are able to add properties onto the nodes). I believe what you are describing is counting the nodes in a subgraph that is disconnected from the rest of the nodes in a Neo4j db? There is at least one algorithm in apoc that can be used to label all nodes in every disconnected graph with a unique group number, once that is complete then a cypher to count the nodes in a group (all with the same group number) is quick and easy.
More information on a related thread
Doing this can be very useful to get a perspective on what is in a graph, the characteristics and number of disconnected subgraphs can provide insights.