Chouby on "[Plugin: Polylang] Home page shows posts in both languages / IIS"

That's very strange and although I never tested it, I have no idea why IIS should provoque this.

Could you add these lines

echo '<!--';
print_r($query->query);
print_r(is_home());
echo '-->';

in the file polylang/include/core.php at line 417, just above:

// homepage is requested, let's set the language

It won't solve your issue but may help me.