What is incumbent upon one who kissed and embraced a girl during the daytime in Ramadan until madhy was emitted, and then repented from that? Is expiation (kaffarah) required of him, or only making up the day (qadaa')?

The emission of madhy (pre-seminal fluid) does not invalidate fasting, according to the preponderant opinion. However, the emission of many (semen) due to kissing or similar acts does invalidate fasting. If you are uncertain whether what you saw was many or madhy, then you are not required to make up the fast, and your fast is valid, because the default assumption is the validity of the fast until something definitively proven to invalidate it.