How can I extract all mailbox names and email addresses in my entire organization?

How do I get a list of all mailboxes in my Exchange organization, with their SMTP addresses?

The most appropriate report template to use would be Selected Attributes From Mailboxes, that you will find on the Templates > Inventory menu.

You would select all mailboxes in the organization, select at least three attributes (First name, Last name, Email address) and export your report as a CSV file, and then open that file in Excel.
You would obtain a report such as this one:

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