Is expiation (Kaffara) and making up for the fast (Qada') required for someone who had intercourse with his wife after the Fajr Adhan, forgetting, then remembered after ejaculation, knowing that he completed his fasting that day?

The criterion for the obligation to abstain from things that invalidate the fast is the certainty of the advent of dawn (Fajr). If you continued to caress your wife after the Fajr adhan (definition) out of forgetfulness, then there is no make-up fast due on you, nor any sin, and your fast is valid. However, if you intentionally caressed her after dawn, while remembering (that you are fasting), until semen was discharged, then your fast is invalidated, and you must make it up. The majority of scholars hold that no expiation (kaffarah (definition)) is required if intercourse did not occur.