Your child sounds normal to me. Once they can crawl and walk they want to MOVE all the time. My daughter did the same thing and my son, especially, though he's only 10 months old, is extremely squirmy. I almost can't change him on the changing table anymore because he just wants to sit up or roll off. We don't call him "busy body" for nothing.
As for your trip, it will be challenging. To make the most of it...take LOTS and LOTS of distractions. Different kinds of new toys, games, books and comfort items from home too. At 13 months, she might be a little young, but you could try a portable DVD player and a stack of baby DVD's. Try mini "goodie bags" or "treasure bags" with different "surprises" inside to keep things interesting for her.
Every child is different. It's okay to check with your pediatrician, but you probably don't have anything to worry about. You just have the challenge of keeping your baby safe and entertained.
My daughter was 22 months when I traveled on an airplane with her. We used her car seat the whole way and she was content and comfortable because it was something she was use to traveling in. If you're planning to use her car seat, that might help.
As for the eye-contact thing. That sounds normal too. At that age, they are more interested in looking past you to the next thing they can "get into" than in looking at you or other familiar things. Babies have one track minds. They have really good focus when it comes to investigating NEW things and trying to figure them out. I try to remember this when my son is reaching for my necklace... I say no and start to think about something else, but he hasn't forgotten the necklace and reaches for it again... that's just how they are.