N.O.
Hi N.
Have you bought a seat for your son, or will he be a lap child? If you've bought a seat then you should be able to take the car seat on board and you should strap him into it for the flight. You can check the stroller at the gate. (Usually... it's probably best to call Southwest and get their official policies.) However, if he is a lap child I don't think you can check both a car seat and a stroller at the gate. If the car seat was in a snap 'n go type thing then you could because they'd consider it a 2-piece stroller. (We just flew internationally like that with our 5 month old.)
Alternatively you could check your car seat along with your luggage - but I'd try and pad/protect it somehow, and have it marked fragile.
You can take milk on the flight, just declare it as you're passing through the checkpoint. Shouldn't be a problem at all.
A suggestion for you during ascent and descent - have something your child can suck on during that time as it will alleviate the air pressure. I breastfed my baby during those times on our recent travels and he never cried once. However, we could hear the pained cries of other children all over the plane. :( I don't know if he uses a pacifier, or give him the bottle or something. It truly does help.
Re bedtime, do you have any bedtime rituals that you can include in a small enclosed space like a plane :)? Singing a lullaby softly under your breath? Goodnight prayers? The lights may be on for a while, so maybe purchase him a soft comfy eye mask (pretend it's a new toy or game)? A bedtime lovey/blankie?
I'd also recommend the site: www.jetwithkids.com. Lots of handy advice for all ages on that site.
Good luck and enjoy your vacation! I also had tons of anxiety about flying (30 hour trip door to door from Chicago to South Africa) but it all worked out okay.