Does Elasticsearch capture audit logs for Query DSL, EQL and SQL or Not?

No. The list of events which could be captured by the audit logs is here: Audit events | Elasticsearch Guide [8.9] | Elastic

But you can use slow logs for this: Slow Log | Elasticsearch Guide [8.9] | Elastic. I think that using 0 as the threshold will log everything. But be careful as it will be very verbose.