I think some of the confusion is coming from your verbiage. The words you use for the child's seat when talking to the people from SW will matter, and the answer will change depending on the words you use, so hopefully this will help:
- A lap baby is a baby WITHOUT a seat on the plane. On domestic flights, they fly for free. On international flights, they are charged a small fee (usually a percentage of the accompanying adult). The "bed" they have reserved for the baby will just hook into the bulkhead wall in front of them - they still won't have a seat for the baby flying over. (Personally, I'd slit my wrists before flying with a lap baby internationally, even with the bulkhead bed, but that's just me...) With a lap baby, you don't bring the carseat into the actual plane cabin - it gets gate checked or checked at the counter. If you call southwest and ask about a lap baby, they will assume you are talking about a child under 2 who does not have a seat on the airplane, and will answer accordingly.
- From Orlando to your house, you will be buying the baby a ticket. The baby then becomes just any old ticketed passenger, for the most part. SW does offer "infant fares" as a courtesy, but, as a PP said, they aren't guaranteed to be cheaper than the cheapest available adult fare, as they are a discount off of a FULL FARE ticket - and those are usually hundreds of dollars more than the restricted fares. Assuming you buy the restricted, cheaper, fare, than southwest doesn't care one whit how old the person in the seat is. You can just bring the caresat for the baby onto the plane and gate check the 3 year olds seat and you wan't have any issues at all.
As for flying down with a carseat and without a baby... for us, on Korean Air, when we did that it did count as one of our pieces of luggage. That said, there have been many times when I've asked for a gate check tag for my stroller or carseat when DH and DS are off somewhere else and the kid isn't around. The gate agents usually need to see the actual item (stroller or carseat) but I haven't ever run into anyone asking to see the actual kid. I'm sure if you just don't mention it and bring the carseat to the gate you'll be fine.