I fly Southwest frequently. I love this airline. I've encountered delays, but overall their track record is one of the best.
First of all, the risk of not sitting together will ONLY come into play if the flight is full or nearly full.
It does depend on the airport a little, but unless you are already planning on staying up past midnight, I wouldn't bother to do it just to check-in online. If you do it any time before you leave for the airport, I almost guarantee you'll still get group "A." However, if you have to check any bags, they will redo your boarding passes at the airport anyhow. If you plan on checking bags, just get to the airport about 1 and 1/2 hours ahead of your departure.
Are you planning to park your car? If you plan to park at an economy lot, where you take a shuttle to the terminal, I'd allow an extra half hour for that too - but again it depends on the airport. Also, some airports have a shuttle thing within the airport, but smaller airports don't usually have this. (MCO, my favorite airport - has a fun monorail to get to the main area) I don't mind waiting a little while in an airport over stressing about missing the plane, but then there is lots of traffic between me and the nearest airport.
Having some sort of ID for the kids is always a good idea when you travel, but I think you only need it for Southwest if your child is a lap child (under two, and flying for free, sitting in your lap). I suggest always having a picture of your child on hand too. For a kid's ID, an immunization record qualifies as an ID. I've never seen them check for ID when it comes to being in the preboarding group, but they get a little strict when it comes to Orlando flights with lots of kids.
Worst case scenario - if everything goes wrong and you are the last people to board a full flight - politely tell the flight attendant your situation, and they will ask if someone is willing to move to help you. I've seen it done before, and someone was willing to move. Trust me, most people would rather move to an aisle seat than hear a six year old screaming all the way to Orlando. There are enough adults out there who have a mild fear of flying themselves that they will understand that a child might be afraid too.
Ina side note, you are allowed to bring your stroller right up to the plane. It is called a gate check. Gate checking two strollers is a probably more than you would like to be bothered with, but if you want to use one while you are in the airport, so your kids don't have to walk/don't get lost - it can be done.
Last, Southwest has recently started a new policy about bags not weighing over 50 pounds - or they charge - and I think they only allow 2 checked bags per person. My regular sized suitcase was around 40 pounds - our carry-on sized bags were about 25 pounds. I don't think they counted car seats as limited items. Just know it might be better to have two smaller sized bags than one huge one.