I have about 20 solo plane trips with baby (3mos-21mos) under my belt.
Be ready at the security checkpoint. Liquids need to be out and declared. Stroller needs to go through the machine no matter what. Even the baby has to take shoes off. Baby cannot stay in sling.
Stroller pro: You can use it to carry diaper bag and carry on if not baby and leisurely stroll down to check it at the gate.
Stroller con: It is always a drag to get it through the machine all by yourself. Sometimes they make you wait a long time to get it back from under the plane.
I personaly prefer to travel alone with the back carrier for toddler, a diaper bag on my shoulder, and one roller carry-on. I have learned to pack light and squeeze it all in.
I grab special snack in the terminal for the plane ride. My baby was always more interested in snacks than anything else. I pack some fave snacks and water bottle.
Pack a change of clothes for babe and a top for YOURSELF. It hasn't happened to me yet, but I watched a mom get covered in toddler vomit on the plane.
You know your baby best. Bring what they are interested in. My DD likes to read. I brought a couple of her books that had a lot of open endedness to them. We spent many hours reading sky mall, pointing out the cats and dogs and babies.
I have had very little success with buying/wrapping a bunch of little toys for a trip. That bought me maybe a minute. However, DD could spend a half hour taking the cap off of a water bottle.
Everything is interesting. Everything. Follow their lead. They can spend hours playing with the buckle, the shade, the plastic cup, the puke bag, the air blower, looking back at all the passengers, making a trip to the bathroom, etc.
Make a game or song out of everything. Run through your entire repetoir. It is exhausting. Just remember every minute that goes by is one less minute left.
Bring their fave blankie/lovey and just pray that they fall asleep. Jiggle, pat, sing, (nurse if you can). Then you can relax.
Remember noone is more annoyed at your screaming child than you are. They can put headphones on. As long as your trying your best noone has any right to be mad at you are baby. I have never had a problem (but I do have an angel.)
As far as Southwest. You will get to boared after A group, (unless you are A group) and will get a good choice of seats. Hopefully your plane is not full. I am guilty of trying to make my toddler look worse than she really is during those few minutes of boarding.

I have gotten whole rows to myself. Nobody wants to sit next to a toddler.
As far as other airlines with asigned seating. make freinds with the gate agent right away. Just go up as soon as they get to your gate and ask if the flight is full. They usually take the hint to move you around so you get an empty seat by you.
I know I am missing something. Good luck!