Are we allowed to take batteries on the airplanes? like double a? and digital cameras?

Yeah, you can bring double A batteries and digital cameras on board. I have done it before and i did it recently on a trip.

Yes. You can take batteries. Digital cameras can go through the x-ray. There is no film in it. Also, any film under 800 speed can go through the x-ray.

Yes you can bring batteries on the aircraft with you, as well as your digital camera. The TSA actually suggests that you pack your digital camera in your carry on luggage as it is less apt to get damaged that way during your travels.
Hope this helps and have a great flight!