Is a person who performed Umrah in Shawwal then performed Hajj from his home country in the same year obligated to offer a Hady (sacrificial animal), and what is the ruling on his Hajj if he does not slaughter it, and what is required of him?

The sacrificial animal (hady) is not obligatory for someone who performed Umrah in Shawwal then traveled back to their country. It is only obligatory for the Mutamatti' (one performing Umrah then Hajj in the same journey) and the Qaarin (one combining Umrah and Hajj intentions). It is better for someone in the aforementioned situation to perform Hajj as an Ifrad (Hajj only).