Is spending the night outside the home in the a car until dawn considered fulfilling the husband's oath: "You are divorced by three if you don't spend the night in the house today"? And is another expiation required?

The husband's statement: "By triple divorce, you will not spend the night in this house today" is an oath by divorce. The majority of jurists hold that divorce occurs upon breaking such an oath.

However, some scholars, including Ibn Taymiyyah, are of the view that this oath refers to the intention of the speaker. If he intended what is typically intended by an oath (encouragement or prevention) and did not intend divorce, then its ruling is that of an oath, and he must pay an expiation for an oath (kaffarat yamin) upon breaking it, and no divorce occurs. But if he intended the occurrence of divorce, then his wife is divorced upon breaking the oath. This opinion is the more preponderant (rajih).

In the event of divorce occurring, only one divorce takes place, because triple divorce is forbidden, and only one divorce occurs from it, according to the preponderant opinion.

If the husband intended, by the wife not spending the night, her leaving the house to any destination, and she left as you mentioned until sunrise, then he has not broken his oath, and nothing is binding on him because what he intended has been fulfilled.

The husband should fear Allah and avoid using divorce during disputes.