Explore the logic gates used in a basic arithmetic circuit for binary addition. You are given two $2-$bit binary inputs, A and B$,$ and your task is to design a combinational logic circuit that performs the addition operation $($A $+$ B$)$ and produces the $2-$bit result C$.$

$1.$ Utilize logic gates such as AND, OR$,$ and XOR gates in your design.

$2.$ Present the truth table and circuit diagram for this arithmetic circuit to illustrate its functioning.

$3.$ Provide a step$-$by$-$step explanation of how the inputs are processed through the gates to obtain the final output.

$4.$ Furthermore, conduct an analysis of the circuit's behavior when different inputs are applied, and categorize the possible outcomes based on the rules of binary arithmetic.