Is his fasting considered showing off if he proudly answers "Yes" when asked "Have you had breakfast?", then regrets it and completes his fast sincerely for Allah? And does showing off nullify a deed if done forgetfully?

If a fasting person is asked after completing their fast, "Did you break your fast?" and they reply, "Yes," boastfully, then if their boastfulness is out of joy for obeying Allah, there is no harm, for the Prophet, peace and blessings be upon him, said: "Whoever is pleased by his good deed and displeased by his bad deed, that is a believer." However, if they intended to show off their deed to people or to admire it, then it is blameworthy. Scholars have differed on whether showing off (riya') that occurs after an act of worship invalidates it or not. As for riya', it does not occur out of forgetfulness, but rather by the person's will. If one commits it out of ignorance, there is no sin upon them if they are excused.