from my experience in flying you usually get to carry 1 suitecase (carry-on size) and a camera bag (shoulder type), breifcase or laptop bag. I have never been on a flight that allowed animals in the cabin..... I wish you luck in that. I have been carrying my camera gear in a lowpro Nova AW1 bag as the second carry on bag for the past 2 years and never had an issue. It goes under the street in front of me.
Call your airline and ask them about the requirements as many "especially" international have even more restrictive requirements.
Let us know what they tell you.
As for carrying your gear carry it on to the plane if at all possible. if it means the backpack is gonna be too big get a smaller bag and carry on what tyou can and but the rest in your other luggage (i.e. batteries, memory cards etc)