What is the ruling of Islamic law concerning a husband who wears his wife's underwear and uses her makeup, engages in homosexual relations, watches perverted films, and practices anal penetration, especially given that the wife can no longer tolerate dealing with him in the marital relationship, and how can a separation be sought while preserving the reputation of the children?

These actions are ugly, and contrary to Islamic law, human nature, and chivalry. It is not permissible to obey your husband in them. You should be frank with your husband, advise him to repent, and suggest he see a psychiatrist. If he repents and becomes righteous, then live with him in kindness. But if he persists in his deviance, it is better for you to separate from him through divorce or khulʿ, and conceal his faults. If he does not divorce you or agree to khulʿ, then you may refer the matter to a judge and inform him of the abominable acts your husband is committing. As for family and relatives, you may inform them of the bad marital relationship without going into details.