What is the ruling on a person smoking his cigarette at the time of the Adhan? Does he bear sin as he would for smoking at times other than the Adhan? And is it incumbent upon the son to be angry and feel shame towards his father if he does that?

Your father should always avoid smoking, especially when reciting the Qur'an or hearing the Adhan. You should advise him gently and kindly, explaining the harms of smoking and inviting him to quit it and repent from it, without harshness or severity.