Do the free templates require Elementor Pro?

No, you do not need Elementor Pro for the free templates provided with the free version of Analog Templates. All the templates that are provided for free are only using the Contact Form widget (which is an Elementor Pro feature). But you can always replace with your favorite contact form plugin.

When importing a free template with a contact form widget, the widget will simply not be imported, but everything else will work just fine.