What should one do who vowed a sum of money if he succeeded in something, and he did succeed, but he does not remember whether he uttered the vow or said it to himself? And what is the ruling on swearing an oath for the sake of honoring someone, such as when one offers to pray in congregation or upon leaving the mosque, and he does not do so, and is this considered an oath that requires expiation?

A vow is not made binding by intention or inner thoughts, but by uttering it. Doubting whether it was uttered does not obligate fulfillment, because the default state is the absolution of one's responsibility. As for an oath, if it is uttered unintentionally, it is considered a 'laghw' (unintentional, vain oath) for which there is no expiation. If you intended to show honor (by the oath), the majority of jurists do not require expiation if the other party does not comply. However, if you intended to compel (by the oath) and the other party did not comply, then an expiation for an oath becomes binding upon you. Some scholars have differentiated between swearing an oath upon someone whose obedience is expected, and someone whose obedience is not expected; expiation is binding in the second case but not the first.