On quadratic residue and non-residue of integers

Question

Let $m,n$ be coprime integers with factorization known. Let $x\in\Bbb N$ be a quadratic non-residue modulo $m$. Let $y\in\Bbb N$ be a quadratic residue modulo $m$.

1. When is $x$ a quadratic residue modulo $mn$? How small can $n$ be given $m$ and $x$ are fixed?
2. When is $y$ a quadratic non-residue modulo $mn$? How small can $n$ be given $m$ and $y$ are fixed?
3. What if $m$ and $n$ are not coprime?

I also want to use Legendre-Kronecker symbol to find quadratic residue modulo $mn$.