Can addition be carried out in less than depth 5?
According to Theorem 3.1 in Alexis Maciel and Denis Therien Threshold Circuits of Small Majority-Depth there is indeed a depth-3 circuit for computing the addition of two numbers. 

The precise bound is $\Delta_2 \cdot \mathsf{NC}^0_1$ where $\Delta_2 = \Sigma_2 \cap \Pi_2$ are problems which have depth-2 $\mathsf{AC}^0$ circuits with both $\vee,\wedge$ gates at top and $\mathsf{NC}^0_1$ circuits are $\mathsf{NC}^0$ circuits of depth one (see the paper for a detailed explanation of the notation).

The main proof ideas are:


First, express the Carry-lookahead circuit as $\mathsf{NC}^0\cdot\Delta_2\cdot\mathsf{NC}^0$
Next, invoke closure properties of $\Delta_2$ to write this as $\Delta_2\cdot\mathsf{NC}^0$
Finally, use the fact (also proved in the paper) that $\mathsf{NC}^0 \subset \Delta_1\cdot\mathsf{NC}^0_1$