Using information from Andy Lester, I was able to come up with the following queries to get the information I needed.
Get tables that include foreign keys:
SELECT cl2.relname AS ref_table FROM pg_constraint as co JOIN pg_class AS cl1 ON co.conrelid=cl1.oid JOIN pg_class AS cl2 ON co.confrelid=cl2.oid WHERE co.contype='f' AND cl1.relname='TABLENAME' ORDER BY cl2.relname;
Get tables that view or rules from the table relate to:
SELECT cl_d.relname AS ref_table FROM pg_rewrite AS r JOIN pg_class AS cl_r ON r.ev_class=cl_r.oid JOIN pg_depend AS d ON r.oid=d.objid JOIN pg_class AS cl_d ON d.refobjid=cl_d.oid WHERE cl_d.relkind IN ('r','v') AND cl_r.relname='TABLENAME' GROUP BY cl_d.relname ORDER BY cl_d.relname;
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