What is the ruling on a husband and wife going to regular restaurants and eating there, and is that considered mixing (ikhtilat) if each of them sits at a separate table?

It is permissible to go to a restaurant if it has private family sections; otherwise, it is not permissible. The prohibition is not due to the gathering of men and women under one roof, but rather because the woman is compelled to uncover her face and relax while eating, among other things that are impermissible for non-mahram men to see of her.