Is there a way to extract the Address External ID? I was able to assign it via CSV import but it does not seem available in saved search, dataset and suiteql.

I think no. For addresses you're better off using the Address ID, address label or Internal ID. IIRC, I don't even think the external ID can even be used as a unique key for imports/updates, so it's pretty useless. But it's been a few years since I was knee-deep in address stuff, so it's possible I'm wrong/someone else knows better.

Thank you.