Lots of airlines have discounted tickets for children under 2, though not Southwest. I've never heard of that money back program, and no one's ever mentioned it, but it's worth asking!
I've flown with my kids many times, and have always bought them seats. (Granted, I am often the only one on the plane with a ticketed infant.) On the routes I fly, there are rarely open seats, and I am just not willing to take the chance. I figure that if the FAA wants me buckled in, then my kids are going to be buckled too.
We have used Snugrides, Roundabouts (front and rearfacing), and Marathons on planes. Depending on the plane, you may have to install it with the armrest up, and it's possible that a bigger seat wouldn't fit well in a puddle jumper plane. Now that my older child is over 40 lbs, we don't bring a restraint on board for her, but bring our Safeguard Go for the rental car at the other end.