If someone swears by "By God, it's a disgrace" and I nod in agreement, do I owe expiation for an oath (kaffarat yamin) if what he swore by is not disgraceful? And do I owe expiation for an oath if I use "yes" or "no" as an answer to someone who swears by God, intending to disclose the correct information contrary to what he swore by, while fearing bearing the expiation without the one who swore?

You are not obligated to offer expiation merely by moving your head, as a gesture is not an oath. Nor is saying "yes" or "no" an oath, because you did not swear by the name of Allah or His attributes. Furthermore, the swearer himself is not obligated to offer expiation if the oath slipped from his tongue unintentionally, or if he believed himself truthful and it later proved otherwise. The feeling of fatigue you experience is due to whisperings (waswasah), so you should ignore them. Three conditions are required for expiation to be obligatory: that the oath be binding, that one swears voluntarily, and that one breaks the oath voluntarily and rememberingly.