Is this considered a divorce or a khulʿ, and how long is the waiting period, and is it necessary for the divorced woman to remain in the marital home during the waiting period?

If the court rules a divorce due to harm with a revocable divorce, it is a divorce, not a khul‘ (dissolution of marriage initiated by the wife). The general rule is that the divorced woman should observe her 'iddah (waiting period) in her husband's home, as Allah Almighty says: (Do not expel them from their homes, nor should they leave, unless they commit a clear immorality). This is due to the wisdom behind her remaining in the marital home, as Allah Almighty says: (Perhaps Allah will bring about thereafter some new matter). However, if there is a real fear of harm from her remaining, such as fear of beating or abuse, then it is permissible for her to move and observe her 'iddah in her family's home.