Queries to see how many content types, and how much content in each

Drupal 7:

SELECT nt.type, COUNT(n.nid) AS node_count, COUNT(CASE WHEN n.status = 0 THEN 1 ELSE NULL END) AS unpublished_count
FROM node_type AS nt
LEFT JOIN node AS n ON n.type = nt.type
GROUP BY nt.type
ORDER BY node_count DESC;

… wonder how hard to get the number of fields on each, also.

Well, here is a full listing of fields per content type from a D7 site:

SELECT entity_type, bundle, field_id, field_name FROM field_config_instance ORDER BY entity_type, bundle, field_name;