I just went to LAX with my 2.5 yr old and my 8 month old earlier this month.
You can bring the carseats on the plane if you, but you may feel that it is a hassle to lug them all the way through security and the airport, although it is OK to do so, and is safer for them to be in one. That said, we didn't bring the car seat on the plane and our 2.5 yr old was fine - just had to tell her a lot of times not to take off the seatbelt when she was sitting there. But, we also made that decision as we didn't need a carseat when we got to LAX, since we were borrowing one there. If you do decide to lug them, maybe get a roller bag or backpack bag designed exactly for this purpose? The are a few cheap ones on amazon, etc.
As for the stroller, if you have a double you might want to bring that, and do baby bjorn with the baby too - that is an option. They do have you break down the stroller and put it through the security scanner thing. Note: strollers & car seats will not be charged the checked luggage fee, regardless if you check them at the gate or ahead of time.
For the kids, if you have a portable DVD player that helped us a lot, plus lots of favorite snacks - (goldfish, etc). I got my 2.5 yr old a backpack and filled it with 3 books, 2 dvds, snacks, and stickers and we had more than enough things to keep her occupied. She still takes a nap from 1-3, so we did an early flight (9am) and I liked it because we arrived just before noon, and we got to our destination about 1:00 and she could still have her nap. In order to make the 9am flight, we got nearly everything in the car the night before and just got them up, put them in their clothes, brought breakfast in the car and went.
As for the air pressure, it did not effect the 8 month old as much as I was anticipating. I tried to nurse her on take-off and landing, but couldn't time it right. She ate while we taxied on the run-way and finished just as we took off, then wasn't hungry for the landing. I gave her a pacifier and crossed my fingers - it went fine.
One thing you may want to ask about if you bring the car seats on the plane, is seat placement. I know that if you have one car seat, it MUST be in the window seat so that it doesn't obstruct the exiting of any passengers in an emergency. With twins, I am not sure if you could put them next to each other or not. You may want to call Alaska about that one.
Hope that helps!