You have multiple questions...I have multiple answers.
First, I used to fly a lot. About 5 years ago (so not too much of a change in airline procedure and airport security) I was averaging between 25-35 domestic flights a year plus some international ones. I never had a problem flying with my gear as carry-on. I usually had a soft-sided carry-on duffle which I managed to get a small, portable soft box and small scrim or backdrop plus other gear in to (as well as clothes). I usually had a big backpack (that carried 1-2 bodies, all of my lens, my laptop, batteries, etc.). I've never had a problem with loose batteries as long as I carried them on. I usually carried a tripod of some sort (though it collapsed small enough to fit in to carry-on) and often times it was just a small table-top thingie or a clamp or gorillapod. If you shoot landscape and you're going in to big sky country, you should bring something to stabilize your tripod otherwise you're going to miss a helluva sunset.
Second, shooting at airports...it's against the regulations to shoot pictures in the screening area so make sure your camera is put away. No problem with x-ray screening of digital equipment. But I take a lot of pictures at airports. Don't focus on security stuff (door locks, security personnel, k-9 teams, etc.) and you should be fine.
Third, taking pictures on airplanes? Bleh. I've seen some lovely ones. But most windows I've seen are badly scratched, have water drops on them. Maybe you'll get lucky. Also, check to see what kind of plane you're flying and where your seat it--odds are most or all of your view will have a wing or engine assembly obscuring the view.
Fourth, here is the biggest concern I have for you...it's sounds like you're not a frequent flier. It also sounds (I'm guessing) like you're not flying first-class but coach. Check your boarding assignment (you probably can't until the day you check-in, usually no earlier than the day before the flight). You are probably not boarding in sections #1 or #2. Which means you are running the risk that you will NOT be allowed any carry-on bags (except something that fits under your seat). Basically what I'm saying is that the overhead space fills up quickly and the people who board later on the flight effectively are only allowed one carry-on. If you're in section #5 or #6, that's pretty much a guarantee. So pack accordingly--get all of your camera gear in to one bag that you can fit under your seat and allow for your feet as well (you'll also need to get key toiletries, medicine, etc. in there as well). Otherwise, you're going to be checking camera gear. And I never, under any circumstances, check camera gear. I always do carry-on. Or if it's too much for carry-on, then I shop separately, in a pelican case, with insurance, for someone who is a shipper (like FedEx or UPS) rather than an airline.
Fifth, travel tips....well, you're going to spend time waiting. So I have Sudoku on my phone. Or bring a cheap novel you don't mind tossing when you're done. Or print out a couple of articles on photography you've always been meaning to read. I have Nikon Learn & Explore on my phone...lots of quick reads I can easily access. A rule that all frequent travelers follow if it's not the same trip back and forth ever time is...bring some food. Doesn't have to be a lot...a bag of granola or trail mix. You may get some cheap pretzels on the plane but probably not. Bring an empty water bottle through security and then fill it up when you pass through. You'll get a beverage on the plane but until then you'll end up buying stuff in the terminal (very expensive). Most airports have sculptures of various types or mobiles hanging from the ceiling. So pull out your camera and shoot some abstracts (especially play with DoF). If you've got a long enough zoom (so you're not intruding in people's space) there are some great people shots (basically street photography except in an airport terminal). If you've got a long connection, it's worth it to have a pass for an airline lounge...reliable wifi (most airports are still crap regarding public wifi), lots of free beverage and snacks, some magazines, comfortable seating, not as noisy as the terminal. Bring a set of ear buds. You'll want to listen to music (even if it's on your laptop or phone) or the headset on the plane will be broken and there's a crying infant in the row behind you. Finally, get up and move a lot. It's healthy for you, avoids cramps and reduces jet lag. For longer flights, I will often download a movie from iTunes to my laptop. Even on the plane--if the flight is 2 hours or longer, every 30 minutes just get up and walk up and down a aisle a couple of times then so stretch in the back and head back to your seat.