Yes, when you create a new Popup, you can link it with an existing discount code in your store. The popup inherits the same configuration of the discount, for example: if your discount targets collection A, the Popup will be shown only if user is exiting from a page of product of collection A.
Just link the Popup with a discount targeting all products, the Popup will then be shown on all pages including non product pages such as home.
Yes, in the popup configuration, you can choose to hide the discount code and request the user to enter his Email address before showing the discount. All Email addresses are saved to your Shopify customers list under the tag of ‘leads’.