Does an expiation become necessary for an oath that was broken? And does what was issued by the husband in the other two situations count as a divorce, and what are the resulting rulings concerning that?

In the first scenario, no expiation is required because the oath is dissolved. As for the second scenario, if the husband did not firmly intend divorce, it does not occur upon the fulfillment of the condition, because the default state is the continuation of the marriage. Regarding the third scenario, if the wife browses her husband's phone, divorce occurs, whether he permitted her or not, unless his intention was merely to prevent her from browsing conversations, in which case he would not be breaking his oath by her opening other things, because the intention of the one swearing an oath specifies the general.