If I have to prove that $A_{kh}=g_{ij}\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}$ is symmetric in $k$ and $h$, knowing that $g_{ij}=g_{ji}$, for the term $\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}$ it is allowed to rename $i$ with $j$ and viceversa and so can I write the following? $$g_{ij}\frac{\partial x^i}{\partial y^k}\frac{\partial x^j}{\partial y^h}=g_{ij}\frac{\partial x^j}{\partial y^k}\frac{\partial x^i}{\partial y^h}$$ I am not so sure that this is only a matter of renaming the indeces, since for instance if I rename the indeces for a very general (no symmetric) matrix $s_{kj}$ (I rename $k$ with $j$ and viceversa) this becomes $s_{jk}$ but this would mean that this $s$ is symmetric.
So how can I prove really the symmetry of $A_{kh}$?
You can rename indices that are summed over ($\sum_i f(i) = \sum_j f(j)$), so your derivation of $A_{ij}=A_{ji}$ is correct. One is not allowed to rename so-called free indices, whose value is fixed -- doing that would lead to the wrong conclusion that every matrix is symmetric, as you pointed out.