Flying with a baby is not hard at all. As the others have said, feed or give them a dummy on take off and landing.
Bub needs to have their own seatbelt on, which attaches to yours so the sling may not work once you are seated. I have always used one getting around the airport though...it is somewhere that you need the use of your hands.

Have you actually called budget to check about the car seat or did you do it all online? In my experience they all have convertible seats so infants are fine in them, you just turn it around when you get the car. Probably worth a check rather than take your own seat for no reason.

I second not taking any valuable baby equipment on the plane if you can avoid it. We have a cheap stroller especially for taking on planes, our good one got broken the first flight we took it on. Qantas are generally nicer than the other airlines as far as letting extra baggage slide so you might get lucky.