Can we stack up gift cards or a gift card and a discount code at Checkout?

The answer is Yes, you can. 

1. More than one gift card can be used towards a purchase.
2. Gift cards are a form of payment, and can be used in conjunction with a discount code. Customers should be able to first apply one gift card code, and then another using the gift card/discount field during the Checkout flow.

More FAQs about Gift Cards can be found in this Shopify article.