Is it permissible for a woman to seek divorce from her husband who suffers from depression, beats her, mistreats her, has not spent on her for three years, and whose family has refused to discuss divorce with him, and what would be the waiting period (iddah) in this case?

If the situation is as you described concerning your husband's ill-treatment of you, then this permits you to seek a divorce, and the matter should be referred to the Sharia court for judgment, either by divorce due to harm (talaq al-darar) or khul', based on the evidence presented. The waiting period (iddah) for a divorced woman is three menstrual cycles for those who menstruate, and upon giving birth for a pregnant woman, and three months for those who do not menstruate or have despaired of menstruation. The iddah begins from the date of the divorce or the final court ruling.