I am obviously in the minority, but I have always gotten my kids their own seats. I feel it is the responsible thing to do. Yes, really bad turbulence is rare, but I wouldn't be able to live with myself if I made it through a flight and my child didn't cause I couldn't hold onto them tight enough. If I can't afford to buy them a seat then I don't go. I think it is bizarre that cargo is safely stowed under seats and in overhead bins but precious baby cargo is okay to be sitting on your lap. I'm just sayin..
But your question was really about whether to bring car seats and for the 3 and 5.5 year old that I would say no. What we did with my two oldest 4.5 and 2.5 is get them a CARES, the Child Aviation Restraint System. It is for kids 22-44 pounds. Once they are out of that range, I will just use the belt on the plane. It is kind of expensive, but we've used them enough times now to make it pay for itself in no seat hassles. Your oldest may already be out of that weight range. If you are wondering, the belt is super easy to install -quick and fits in a diaper bag. As far as car seats go. Yes, you can check them usually for free with luggage. But, keep in mind that they may have hundreds of pounds of luggage thrown on top of them, thus making them less safe. I would carry them and check em at the gate. I usually put my youngest and infant seat in a stroller frame and have my other two walk. Then my husband carries the two seats and wears a backpack full of stuff to do/diapers. If needed, I will also have another bag under the stroller to carry on. Yes, it's a lot of stuff, but worth it to me. Hope that this helps you!