Adder hardware logic
This circuit is more complicated than necessary, but it looks like C2 does give the correct output.  The simplified expression for C2 is:
$$
C_2 = \overline{\overline{(A_1+B_1)}+(\overline{C_1} \cdot \overline{A_1B_1})}\\
=(A_1+B_1)\overline{(\overline{C_1}\cdot\overline{A_1B_1})}\\
=(A_1+B_1)(C_1+A_1B_1)\\
=C_1A_1+C_1B_1+A_1A_1B_1+A_1B_1B_1\\
= C_1A_1 + C_1B_1 + A_1B_1
$$