Does the wife become forbidden if the husband says to her, "You are forbidden to me if you leave the house," intending to specify leaving for certain places such as beauty salons and markets, then she leaves for a dire necessity with his prior knowledge of that, and they later reconciled and he allowed her to leave to attend to her needs?

Prohibiting one's wife is ẓihār. Therefore, if one makes the prohibition of his wife conditional on an act, and she performs that act, then he has committed ẓihār against her, and he must offer the expiation for ẓihār before approaching her. What is most important is your intention (niyyah): if you intended a specific place or time, then you do not break your oath if she goes out to a place or at a time other than that. However, if you intended an absolute prohibition, then you break your oath merely by her going out without your permission. Reconciliation does not nullify the oath unless your intention was linked to a state of anger. If she goes out in the manner you intended, you have broken your oath. If you had no specific intention, then the wording is general and remains so, and the oath is broken merely by violating it.