Can the Follow Up Email module send emails to my customers with auto-generating coupon code?
Yes, it allows generating coupons automatically. You have to create the Shopping Cart Price Rule (Promotions > Shopping Cart Price Rule) and select Follow Up Email Generated Coupons value (Auto for Magento Enterprise Edition) in the Coupon field.
After this you may select shopping cart price rules which has Coupon as Follow Up Email Generated Coupons in Shopping Cart Price Rules field at Coupons tab when you're editing or creating Follow Up Email rule. When the rule is triggered new coupon is generated and it will be available in the email template.