There's no charge to book over the phone, unlike many airlines, so that's an option.

I have not used a Groupon, but have used gift cards online more than once:

The groupon code goes, I believe, into the Promo Code box. Click on "advanced search" to get a booking screen which includes the coupon code box.

The gift card you use at checkout -- you choose it as payment type, and then key in gift card number and PIN. If the card does not have enough value, you are then prompted for another payment method to cover the balance.