Concerning homogeneous polynomials: Let $P(x,y)=\sum_{j=0}^n a_j x^j y^{n-j}$ be such a polynomial, of degree $n$, with some of the $a_j$'s nonzero.
If $n$ is odd, then every line through the origin will have at most one point of intersection with $C:=P^{-1}(\{c\})$. So, then $C$ cannot be a simple closed curve -- because every line through any point interior to a simple closed curve must intersect the curve in at least two points.
It remains to consider the case when $n$ is even. Then $C$ is symmetric about the origin, and hence so is the interior of $C$. Then the centroid of the interior is the origin, and it does not depend on the level $c$.