Will my staging site email my customers?

Not with Email Delivery. A copy of a site at another address (a staging, development or local copy, or a clone) holds every message in the log instead of sending it, and so does any site WordPress calls staging, development or local. Nobody receives anything, and the log lists each message under Held.

On a copy, press Keep holding on the notice and it stops asking; the Dashboard still says email is being held. To check the mail settings from a copy, use Send a test, which still goes, and you can send a held message to your own address from its page in the log. If you do want a copy to send everything, choose Settings → On a copy of this site → Send as usual.

That includes a copy made the usual way, by copying the database and then replacing the live domain with the copy's everywhere in it: the plugin keeps the address it was set up at in a form that replacement does not change.

A copy made from a site that was not yet running this version notes its own address when it is first updated, so it is held only if WordPress calls it staging, development or local.