On the static copy there is no WordPress to receive a form, so each submission is sent back to the original site, which emails it to the site owner. The exporter settings cap how many submissions one visitor can send in ten minutes.
Contact Form 7
Check: try sending it empty, then with a broken email address, then properly. A green confirmation appears under the form and the email arrives with Reply-To set to the address you typed.
Plain HTML form
Check: the ticked checkboxes arrive in the email as one line.
Newsletter form that posts to another website
Check: this form is left alone on purpose. Submitting it goes to example.com, the same as on WordPress.
Search
Check: search needs WordPress, so on the static copy it just shows the home page. It is never sent as an email.