Does a divorce issued by the husband in a state of anger, and with the intention of protecting his private belongings, take effect? And what is its expiation, and how can the wife be taken back if it does take effect?

What you uttered is a contingent divorce of your wife upon her sister touching something for a certain purpose. Whenever her sister touches anything you intended, the divorce takes place according to the majority of scholars. You can take her back before her iddah (waiting period) expires if this was the first or second divorce.

Reconciliation is effected by explicitly stating the intention to take her back, or by an action indicating this with the intention of reconciliation, or by sexual intercourse, even without the intention of reconciliation, according to the sounder opinion. There is no expiation for this, and anger is not considered unless it leads to a complete loss of awareness and comprehension.

Some scholars, like Ibn Taymiyyah, are of the view that if one did not intend for the contingent divorce to take effect but rather intended to admonish or threaten, then the divorce does not occur, and an expiation for breaking an oath becomes incumbent upon him. However, the opinion of the majority is stronger and more cautious.