Is a woman permitted to go out to work if no non-mixed work is available, and with her husband's disapproval of her going out, knowing that the husband abuses his wife and sometimes beats her? Is she permitted to seek a divorce in this case?

It is not permissible for a woman to go out to work without her husband's permission, unless he refrains from providing for her. The default rule is to prohibit a woman from working in a place where men and women mix, except out of necessity. It is advised not to seek divorce, but rather to strive to reform the husband with patience, wisdom, and gentleness. Remember that divorce often does not bring good, especially with a pregnancy, and that a child has a right to the care of both parents. One must repel evil with good and patience, and reflect on the words of the Almighty: (And not equal are the good deed and the bad; repulse [evil] with what is better...) and His saying: (And perhaps you dislike a thing and it is good for you...). It is also advised to begin by reforming oneself, as this helps in reforming the husband.