Is it permissible to sit down to eat with one's father, mother, and an unveiled maternal aunt who does not cover herself in front of the father, knowing that this is her custom and she shakes hands with men, and the father sees no objection to that, and does the questioner incur sin by doing so?

Your maternal aunt is a mahram to you. It is permissible for you to sit with her and look at what is permissible to look at of her. However, your father is not a mahram to her, so it is not permissible for her to shake hands with him or to display her adornments before him, unless there is a mahramiyyah between them through breastfeeding. This is a reprehensible act (munkar) that must be changed by hand, by tongue, or by heart. You should advise them with gentleness and wisdom. If you are unable to change the munkar after offering advice, then you have fulfilled your obligation. Do not sit with them at their dining table if they insist on the munkar, except for the necessity of food.