Is it permissible for a woman to annul the engagement or seek divorce before consummating the marriage with a man when it has become apparent to her that his sister and his cousin's son suffer from obsessive-compulsive disorder, and she fears that he might also be afflicted, despite him being kind?

A man's relatives being afflicted with obsessive thoughts (waswas) does not mean he is afflicted with them; the default is that he is sound. Even if his affliction is proven, it is not a defect that justifies the option of فسخ (annulment) unless the waswas reaches the level of harmful insanity. We advise the questioner to strive to ward off intrusive thoughts. If she believes that his condition will harm her and affect the marital relationship (husn al-ʿishra), then seeking a divorce or khul' is permissible due to need, but impermissible without a need. The hadiths that warn women against seeking divorce are understood to apply when there is no legitimate reason for it.