Have you paid for a seat for your child? or are they sitting on your lap. Usually, an infant will be sitting on your lap aand they give you a second seat belt for the child that has a loop to attach to your normal seat belt. Bear in mind when doing this that 2 people must not be within the one selt belt, because in the case of an accident the adult will crush the child.
Not sure about AirAsia, but they might sit you in a bassinet position on the aircraft and your child can sleep in there. This also has a seat belt arrangement, I believe.
If the child has their own seat, I'm not sure what happens. But I can't imagine that there would be a car seat type arrangement.
You should check your carry on baggage allowance to see if you can take a pram on board. But even if you can I'm not sure how you would be able to set it up and use it. There would be no room really.
A Passport should be sufficient proof of age, but again best to check with the airline.