How to determine all $2 \times 2$ normal matrices? 
Determine all $2 \times 2$ normal matrices. 

In particular, how would I show that there are normal matrices which are neither unitary, Hermitian, skew-Hermitian, symmetric, nor skew-symmetric. The only thing I do know is $AA^* = A^*A$, but I'm not sure how to proceed. I tried writing in $a,b,c,d$ as entries of $2 \times 2$ matrix, but it seemed to lead nowhere.

I'll give a slightly different take on the other answer.
Start by noticing that the identity component of $A$ doesn't affect it being normal, that is, if $A=\alpha I+\tilde A$ with $\tilde A$ traceless, then $A$ is normal iff $\tilde A$ is. We can thus assume without loss of generality $A$ to be traceless, and write it as
$$A = \begin{pmatrix} z & b \\ c & -z\end{pmatrix},\qquad z,b,c\in\mathbb C.$$
The normality condition, $AA^\dagger=A^\dagger A$, translates into the following two conditions on the parameters:
$$
\begin{cases}|b|=|c|, \\ c\bar z=z \bar b.\end{cases}
$$
Writing the polar representations of the parameters as $z=r_z e^{i\varphi_z}, b=r_b e^{i\varphi_b}$, $c=r_c e^{i\varphi_c}$.

*

*Assuming $z\neq0$: these conditions amount to $r_b=r_c$
and $\varphi_c +\varphi_b = 2\varphi_z$. It follows that the generic form of a normal 2x2 complex matrix is
$$
A=\alpha I + \begin{pmatrix} r_z e^{i(\varphi_b+\varphi_c)/2} & r_b e^{i\varphi_b}\\ r_b e^{i\varphi_c}& -r_z e^{i(\varphi_b+\varphi_c)/2}\end{pmatrix},
$$
for any $\alpha\in\mathbb C$ and $r_z,r_b,\varphi_b,\varphi_c\in\mathbb R$.
We can further simplify this assuming $\varphi_z=0$ (which amounts to collecting a global phase for the matrix). We then get the general solution
$$
A=\alpha I + e^{i\varphi_z}\begin{pmatrix} r_z & r_b e^{i\varphi_b}\\ r_b e^{-i\varphi_b}& -r_z\end{pmatrix},
$$
for all $\alpha\in\mathbb C$ and $r_z,r_b,\varphi_b,\varphi_z\in\mathbb R$.

*For $z=0$, the only constraint is $|b|=|c|$. The general solution is thus
$$A = \alpha I +\begin{pmatrix}0 & b e^{i\varphi} \\ b & 0\end{pmatrix},$$
for any $\alpha,b\in\mathbb C$ and $\varphi\in\mathbb R$.

