What is the ruling on mixing (between genders), and are there any sayings of scholars that permit it absolutely or forbid it absolutely, with a detailed explanation of these sayings? Is it permissible to eat in mixed restaurants with the family if the mother uncovers her face and hands, given the questioner's belief that it is not permissible? What is the ruling on the questioner sitting with his sister who does not adhere to the Islamic hijab, and does he incur sin for that?

The mixing of men and women in one place is neither absolutely forbidden nor absolutely permissible. Where the gathering adheres to the controls of Sharia, it is permissible. However, if it includes anything that violates Sharia, such as tabarruj (displaying adornment) or speaking in a seductive manner, then it is prohibited.

Regarding your sitting in restaurants with your mother while she uncovers her face and hands – because she believes it is permissible – and you believe it is not permissible, the consideration is given to the conviction of the doer. As long as she believes it is permissible, and there is no clear text that invalidates her ijtihad (independent reasoning), then her action is the one that matters.

As for your sister who displays her adornment to non-mahram men, if you forbid her from doing so and explain the ruling of Sharia to her, then there is no sin upon you.

It must be noted that excessive scrupulosity in questioning and over-analysis of matters is blameworthy in Sharia and results in a waste of time for both the questioner and the one being asked.