I’m a woman who wants a divorce. How do you complete an Islam divorce? What are the steps? Also, how does child custody work between us… He’s verbally, mentally, and physically abusive to me and the children.

In such circumstances, I advise that you visit your local scholar/counselor.

Generally speaking in cases of abuse even if the husband does not consent to divorce, it can be granted from the religious authority representation. That is a matter that is often dealt by local scholars and witnesses.
With regards to children and custody. Generally, the husband is in charge of providing for them after divorce and they stay with their mother until two and as for girls until seven and it’s recommended that both boys and girls stay with the mother until seven. If witnesses and the religious authority establishes that the husband is not fit to care for the children, they could remain with the mother after seven while the husband provides financially.

Nevertheless, it is best to reconcile and work on your marriage. Now that there are children and they are young it is most advisable to sacrifice for their sake, and try and use the means available to better the environment for yourself and for the children. The holy Quran commands that you bring someone from your side, and he brings someone from his side and lay the matters on the table and discuss them. If their intention is reconciliation, God promises that he would make it happen for them.