You seem to have done your research and have a good plan for the travel. Everything you said about baggage seems to fall in line with what is allowed. I would not count on staff to place anything inside a stroller bag for you, if that's what you want to happen you will need to assume that you will be doing it. Yes, talk to the gate agent when you arrive at the gate and request to gate check the stroller. You can also clarify where you will reclaim the items, whether that is at the gate or at baggage claim. This can vary depending on what your needs are. I can't speak to any specific storage location for car seats, but can say that Southwest handles MANY car seats everyday and I have not heard any reports of damage. If you do notice something speak with a Southwest baggage representative right away.
--TheMiddleSeat
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