Is expiation for an oath obligatory upon one who swore not to omit four rak'ahs before and after Dhuhr prayer, then omitted them? Or upon one who swore not to let his companions pay for tea, but they paid? Or upon one who swore he possessed no money, but had $200?

You should avoid excessive swearing. Your oath to pray four rak'ahs before and after Dhuhr is a binding oath. If you fail to perform them, a single expiation (kaffarah) is incumbent upon you, unless you intended repetition or custom dictates it. As for your oath to pay the bill for your friends, if it was a slip of the tongue, then there is no accountability for it. However, if it was intentional, then if the intention was to honor them, there is no kaffarah. But if the intention was to obligate them and they did not comply, then the kaffarah is on the one who swore the oath. The third oath, regarding the two hundred, if it was not in your consideration, then it is a vain oath (laghw). But if you were aware of it, then it is a false oath, for which repentance is required, but no kaffarah. The expiation for an oath is feeding ten poor people, or clothing them, or freeing a slave. Whoever cannot find these, then fasting for three days.