> Is it possible to refine/combine the above two sets of queries into one?

Yes. It's possible:

A)
        SELECT e.to_node_id AS node_id FROM edge e WHERE
e.from_node_id = $node_id
          UNION
        SELECT e.from_node_id AS node_id FROM edge e WHERE
e.to_node_id = $node_id


B)
    SELECT count(edge_id)
    FROM edge
    WHERE
      edge.to_node_id IN (Query_A)
   OR
      edge.from_node_id IN (Query_A)
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