I've traveled many times with littles.
You can take the stroller and car seat all the way to the gate. I'd prefer that you do this anyway. You want to push baby through the airport, and you don't want stroller and car seat put on the plane until last (less chance of damage). Will you use the car seat on the plane? Does baby have a seat? Or are you planning on holding baby? Just don't check the car seat if you plan on using it! :)
ETA: You will need to get a gate pass for your stroller and car seat AT THE GATE when you get to it. Then you will drop them off at the entrance to the plane. Also, be prepared to put stroller and car seat through the security conveyor belt. You will have to hold baby (no blankets!!!) when you walk through the xray machine.
The car seat is meant to be used in the way you are talking. There are many moms who don't have the base. It's not the best way, but it will work.
The shuttle won't care if you even put the baby in the car seat. I'd do it, but be prepared to have them pulling away as you are still buckling. Practice putting in it in the center of your car....the one with the seatbelt that clamps down (no shoulder strap). Use the center seatbelt in the rental too. Also, check on the back of your infant seat to see if there is a gold buckle thing that comes off. This is for clamping down the shoulder seatbelt. Practice this too, if you need to.....more for when you get to the rental car.
Baby food: I know it's a hassle, but I would get the baby food in the pouches. You won't have to worry about it breaking. If you do take bottles, wrap them well, and put in a bag if possible -- to prevent it from leaking on other stuff.
Good luck, have fun!