Mom of 4. We travel cross-country 4-5 times per year, so we have developed a system.
We pull up to curbside check-in and dump all of the luggage. If there is no line, we show both adults id's at the curbside check-in and then DH leaves to park the car. If there is a line or it is cold, I take all of the kids inside and get in line. DH usually makes it back before check-in. If not, we wave the next party through and wait for DH.
We do not take carseats on board the plane. If your 4yo is small, you might want to, but the 7 yo does not need one. You can check the carseats and they will put it in a big bag to protect it. It's not perfect, but it is okay.
We check luggage and head to security.
Each child carries his/her own backpack with plane toys/games/books. They test it ahead of time -- if it is too heavy, they need to eliminate and repack. The kids pack their own stuff (with guidance for the younger ones -- I don't even check anymore for my 9 and 11 yo dd's) We no longer need a stroller, but it was great when my youngest DS had one! We always gate checked it (just ask at the boarding gate for a gate claim-check for a stroller. If you are at the front of the plane, you will probably have to wait for them to bring it up after the flight).
At security, we ready ourselves ..
Crocs or slip-on shoes are definitely helpful. Wear socks, though, b/c no one wants to be barefoot on an airport floor!! Everything goes through the xray (including small strollers). One adult goes through then the kids, then the last adult. You will need a boarding pass, but not an id.
After that, assuming no security check (I have been scanned before with kids who couldn't walk yet -- I was not a happy camper!!!), you are home free. Grab a water bottle and you are good to go!
It's not too bad, but give yourself plenty of time so you aren't stressed about making your flight. My DH likes to make it with just minutes to go. I like about 20 minutes (or more) to grab a magazine and a Coke!