Visible to tagged or untagged customers
Can I set the discount to show even to new customers who have not been tagged yet?
By default, the discounts are actually shown to ALL customers. You can set it otherwise though by setting the app to only show the discounts to customers who have logged in with their accounts.
Can I exclude tagged customers for the discount?
No, this is not possible yet. At this point, the app can only be set to show the offer to specifically-tagged customers and not hide it instead.