Responses
Yes they are seen as an adult, like we have seen for Estelle, she can conduct business and get married but she needs the permission of her father (head of house). Carlo at 20 is supposed to already be engaged as seen in the tea party chapter. I can guess he does business for the dukedom too. Basically Estelle and Carlo represent the family and can speak in its name now that they are of age (at least that's how I understand it, and that's how it is in most manga/manhwa)
In this manhwa, it said that she had to have a ceremony before she could dance at a party? She had a coming-out-of-age ceremony too? (Was that the ceremony she had to take?) also, did her brother have a coming-out-of-age ceremony?