List all triggers:
SELECT trigger_schema, trigger_name FROM information_schema.triggers;
Find tables without auto-increment primary key:
SELECT t.table_schema, t.table_name
FROM information_schema.tables t
WHERE NOT EXISTS
(SELECT * FROM information_schema.columns c
WHERE t.table_schema = c.table_schema
AND t.table_name = c.table_name
AND c.column_key = 'PRI'
AND c.extra = 'auto_increment')