Is it permissible for a sister to separate from her husband civilly (Danish civil divorce) in order to obtain independent housing for herself and her children, while the legitimate marriage contract (nikah) remains valid?

It is the wife's right to have independent housing. If she chooses to forgo this right to help her husband, she will be rewarded for it. As for divorce, if an explicit divorce pronouncement is issued by the husband, it takes effect, whether serious or in jest, based on the Prophet's (peace be upon him) saying: "There are three matters whose seriousness is serious and whose jest is serious: divorce, manumission, and revocation [of divorce]." The wife is advised to be patient in living with her husband's children to earn reward and gain his satisfaction. Furthermore, it is not permissible to toy with the rulings of divorce, as Allah Almighty says: "And do not take the verses of Allah in mockery."