I'd use the car seat for the older one and gate check the other items. I have to strongly second the comfort level and adherence to the "rules" traveling with the car seat. Also, there's more of a comfort zone for the child. Much better sleeper and sitter and less anxious, in my opinion. Plus, you can restrain them. I know that my younger one loves to click and unclick the safety buckles and would be doing that the whole time with her own.
In addition to the reasons above for gate checking, I have to add one. The use of the stroller and infant seat during the 2+ hours your likely to be at the airport is a god send!! (Ditto if you have to change planes.) I'm actually worried, because this next trip we're not using it as my daughter outgrew her car seat and we're not using it. It was the perfect way to keep her comfortable while we stood in lines, if we had to hurry, and if she was walking, the stroller was the "stuff holder".
You don't need any bags or covering, though there can be a concern with staining if you don't have one. You can wrap in garbage bags for protection, particularly for the infant seat. Though, you would want to use the draw string kind, as they will want to open and inspect it. One of the best items I ever bought when DS1 was a baby is a car seat carrier. It's a zippered "bag" the is cut to fit a car seat. It has handles and a shoulder strap, and once you've left the car, you can put the car seat into the carrier and go. We always put the blankie and the "lovie" and other things like that into the car seat holder and toss it in. (When we don't use the car seat on the plane--like when they were too little and we didn't get them a seat--we piled all sorts of stuff into that bag, there's so much free space. We had diapers and formula and all those things that are very important, but didn't need on the plane.) It cost around $70 at the time, but I tell you it has been wonderful not having to figure out how to lug a car seat across an airport or worry about it getting ruined. We've kept it and used it through 3 kids.