
They're Half brothers since they share the same blood of their father, so yeah, they're brothers. Step brothers is when you dotn share any blood relation. In many cultures, including Korea, you don't adopt your siblings since you're already by law related and are part of the same family registry. Adoption is usually of someone who isn't related to you. For example, your aunt can't adopt you but they can claim custody over you as your family. The little Prince will always be taken care of as a member of the Royal family and his brother, the new emperor will be his custodian.

Left alone where? The little prince has an entire wing at the palace where he lives with servants and attendants at his side. He also has extended family from the Queen and King's side, including a brother, who is a crownnprince. He has a guaranteed job as the nation's Prince, he'll probably be a minister serving alongside his brother. And since his older brother is gay, the little Prince or his kids will be next in line to serve as emperor after his older brother passes on.
I HAVE A FEELING TYAT THEY WILL ADOPT THE KID OF THAT BITCH WITCH QUEEN IF THE POOR THING GETS TO BE ALIVE