Hi A.,
As other moms have already said, you can take a stroller to the gate. HOWEVER, ASK when you check in if the stroller will be available to you immediately after deplaning. We have flown NUMEROUS times. My daughters first flight was at 9 weeks old. Granted, these flights were all international flights...but nonetheless ... we have flown a great deal.
Anyway, our stroller is a larger stroller. It was CHECKED at the gate. This meant it was NOT available to us immediately after deplaning. We actually had to pick it up in checked luggage.
I noticed the smaller strollers, like the umbrella strollers were given to people at the deplaning gate. So, please... ASK before you just assume it will be there.
My advice... take as little as possible on the plane. A backpack or a bag you can put over your shoulder is best. It frees up your arms and allows you to keep track of your child when going through the airport.
I would advise the car seat on the plane. Most airlines allow kids up to 2 years old to sit on your lap. However, unless your child is not a typical 2 year old...this is just going to frustrate you. Yes, that means you will have to carry your car seat through the airport. I would suggest you spend the money and get some sort of wheeled carrier for your car seat. It will save your back and arms. It will be worth it! You can resell it at a garage sale, mom 2 mom sale, ebay, or amazon. Traveling moms will snatch it up!
Traveling can be stressful. Do what you need to do to make it as easy as possible.
That being said, don't let your little girl see your fear of flying. It will stress her and you will have more problems. Flying is no biggie and you'll be just fine.
Regarding occupying her on the plane.... take some favorite books, buy 1 or 2 new small books or something that she really likes.... coloring book or something. A new item will keep her attention a little longer then an old one.
I have known other moms to do the benadryl thing. And, that is fine... I'm just not one to drug up myself or my daughter unless absolutely necessary. So, it's your call on that. But, other moms have said it does work.
Popping ears... sippy cup with a straw. The sucking will help eleviate the ear pressure. In addition, tug on her ears. Tug on the top of her ear... up and out. It will help relieve pressure.
I've never seen stroller rentals. Unfortunately.
Really, the best thing you can do is limit your carry on and use wheeled luggage. That is what I found works best.
Have a wonderful trip!