The docs say you can give discounts to specific user groups. However, if a user (who is part of a group that has a discount) is in store and swiping their credit card, how can the system know there is a discount to be applied before the cc swiping occurs? Would the system only recognize the user after they have swiped, which is after they have been told how much to pay?
The customer would have to bring up the discount code that’s provided in the email when the coupon is sent out. That coupon code is entered in to apply the discount. 
Coupons are specific to Square Online, right? I thought Discounts did not actually produce a coupon code and were just applied to specific items or customer groups?
Depending on the how the coupon was created will determine how it’s redeemed. If a customer receives an online store coupon, they can enter in the coupon code at check out on the merchants online store. Otherwise the customer can redeem your coupon by providing the cashier with your coupon code at checkout.