I just realised that the coloring problem is quite trivial, as you can partition the Latin squares in $n-1$ sets, defined by the value of $L(2,1)$ (or any specific coordinates not on the first line) : this forms a partition and two Latin squares in the same set cannot be orthogonal, hence $\chi(G_n) \leq n-1$.

@BrendanMcKay ok thanks for the info. I'm looking at Prof. Wanless website and references in order to get a better understanding of my others questions. Thanks

For future users, just adding here that Hedetniemi's Conjecture has been proven false by Yaroslav Shitov, and there exists graphs $G,H$ with $\chi(G\times H)<\min\{\chi(G),\chi(H)\}$. See arxiv.org/abs/1905.02167