That’s a really smart idea — analyzing past chats to improve your FAQs is a great move for customer experience!
As for exporting Shopify Inbox chats, Shopify doesn’t currently offer a direct “export all” function through the admin. However, one workaround I’ve seen store owners use is accessing the inbox data via connected apps (like Shopify Flow or third-party integrations) or requesting a data export through Shopify Support if needed for compliance or review.