Power adapter and external power

I should give you some information about my configuration:

  • The gprs shield needs 2A only sometimes, when I connect to the network, call and something else.
  • The configuration with a normal power supply I can power all other items, temperature sensor, 2 servos and a camera.

What I want to confirm is that if I am going to attach the battery directly to the Arduino, does it works? I know that Arduino can handle 12 V but I don't know about 7'000mah, I think so that I understood that doesn't care if I have 1000mah o 7000mah, Arduino will take as much as it needs, right?

What about the regulator? If I buy a 12V to 5V at 3A, does it work with the battery? Does it work as I describe the Arduino that it will take max 3A ?

Sorry to be very stupid with all these questions but I want to make sure that I understand everything