Automated queries?

Is there a way to perform automated queries to discover if a .eth domain is available for register or not?

what do you mean by “automated queries”? You can always search the name on http://app.ens.domains/ to see if it’s available or not. If you are a developer trying to automate the process, you could use GraphQL api via https://thegraph.com/explorer/subgraph/ensdomains/ens