Our daughter flew for the first time at 10 weeks old and we got some nasty stares hopping on the plane. However, she did great and no one stared funny at disembarkment.
Lay your baby down in your lap for takeoff and landing - this seems to naturally help with the ears. Also, give her a pacie or bottle during this time to encourage swallowing. (Alternately, you can nurse at these times, but that seems a tough call to me.)
As for all the gear...Leave the carseat base at home. You don't need it. The carseat can be securely attached each time in the rental car without it - check your instructions. Also, check with AA - IF (unlikely these days) there is an open seat on the plane, she can fly in her carseat for free.
You should be able to "gate check" your stroller and carseat (if there's no empty seat). This means you'd take them all the way to the gate and then would tag them such that they'd be the first things out for you to use again on the other end. Now that they charge $15 for the first bag, I'm not SURE about anything anymore, so you might want to call the AA Gold line and check on that.
As for sterilizing, get those wonderful microwave sterilization bags and use them for all the stuff! We fell in love with those things - they can be used multiple times. I believe Target carries the Medela brand of them.
I'm sorry and a bit surprised to learn that Hilton will only RENT you the frig for this purpose. I had to pump during a business trip once and the hotel where we were just provided that when I explained I needed to store breastmilk for my baby. Ask them if the rooms come with a microwave or if there's one in a lobby area (not as convenient, but it'll work).
I agree with others - take your coupons or whatever and buy the diapers, wipes, diaper cream, etc. on the other end. Just pack in the diaper bag extra stuff for the trip and first day.
Good luck and have a great trip!!