The recommendations for use in cars and in aircraft are different. The FAA recommendation for aircraft is that carseats be used by children who weigh less than 40 lbs.; once they pass that weight the degree of protection provided by the lap belt is judged to be equal to that of the carseat. In addition, I can tell you from experience that in coach, any child who is approaching 4 feet tall will have a very difficult time fitting into the space available; depending on where most of their height is, their legs may become trapped between the carseat and the back of the seat in front of them.
Carseats must be placed in window seat positions ONLY (except in a case where you have two carseats in one row, in which case you must place the one that sticks out furthest forward into the window position, and the other in the middle position). Carseats may not be placed in the exit row, nor in the row immediately in front of or immediately behind the exit row. Most carseats also will not fit in the bulkhead rows because those armrests do not lift.
The FAA defines "booster seat" as a carseat that lacks an integral upper-body harness. No seat that lacks it may be used on board an aircraft. Some seats that DO have a harness are also not approved for aircraft use -- check the label.
SWA will only give you an add'l seat free if the flight is considerably undersold all the way to your final destination. If it is sold to more than about 90%, it won't be offered, and your only hope of getting it would be to ask to wait until last to board on the chance that there might be a pair of seats remaining. (So that you can hold the carseat for gate-check until the last minute.) However, they won't ask anyone to move to give a contiguous seat to a lap-child, so long-shot is an understatement. There is no place to stow the seat on board the aircraft; if you don't have a seat for it then you will have to gate-check it.
PS: No parent traveling with a child under 15 may sit in the exit row, because in an emergency that person's first priority would be the children, not the safety duties he/she agreed to take on in return for the exit row seat.