Coupons, promotions (such as free shipping), and gift cards can all be applied in the same transaction.
It should be noted that coupons apply before promotions. This means that if your order total is reduced (by a coupon) below the threshold for the promotion, the promotion will not apply. The total order value after coupons, but before taxes, needs to meet the promotional criteria to qualify.
With that said, gift cards are used as a form of payment and do not reduce the order total like a coupon would. For example, on your order of $60, if you added a coupon for $5 off and had a new order total of $55, you could use a gift card for $20 and pay the remaining $35 on your credit card and still qualify for the free shipping promotion.