If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric? If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?

No you can take the nilpotent matrix 
$$
M=\left(\begin{array}{cccc}
0 & 1 & 0& 0\\
0&0 & 1& 0\\
0& 0&0&1\\
0& 0& 0& 0 
\end{array} \right)
$$
M isn't symmetrical but $A^4=0_4$ symmetric
