Given an arbitrary production org, how can I find out who the Salesforce Account Executive (AE) is and what their contact details are?
The details are exposed via the Checkout that is accessible via the App Selector just before the AppExchange option.
If you can’t access the Checkout ensure that your profile has either the Manage Billing permission or the Checkout Enabled permission. See What is Checkout & how can I access it?
This will take you to https://store.salesforce.com/home where the AE details appear on the bottom right of the page. The Send Mail button is the mailto: link for their email address.