(1) It is the ignorance and folly of the wife. A wife should obey her husband and should stay where he keeps her. It is her injustice to demand for separate house.
(2) She has no right to demand khula in this case.
(3) If she left without your permission and you gave talaq then she shall not be entitled to receive the iddah expenses. Iddah is three haiz (menstruation cycle), it is not four months. However, mahr also shall be wajib after talaq in each case.
(4) If she is a girl then she shall remain in the custody of her mother up to 9 years of her age and if it is a boy then he shall be in the custody of mother up to 7 years of his age. After that the father can take them in his custody. The father shall be responsible for their expenses.
(5) If she sought khula in this case and you granted it then it shall take place and the woman shall become divorced. But she shall be sinful for her unlawful demand.
Allah knows Best!
Darul Uloom Deoband