Is the guardianship of a brother over his sister solely for marriage, and is it permissible for him to force her to leave her mixed-gender job despite her commitment, and does he have the right to prevent her from buying an independent house to live in with their mother?

Your sister's statement about guardianship is correct. The questioner only has the guardianship to marry off his sister, but he does not have guardianship over her money or the right to restrict her actions.

It is stated in the Kuwaiti Fiqh Encyclopedia: "Among the reasons for guardianship over a person is womanhood, without being linked to youth or any mental infirmity, due to His Almighty's saying: {Men are maintainers of women}. This guardianship is confined to two matters: marrying off a woman, and disciplining a wife in case of disobedience."

Therefore, the questioner does not have the right to prevent his sister from engaging in permissible work that does not involve violating Islamic law, nor does he have the right to prevent her from buying a house and living in it with her mother. However, he must offer her advice, enjoin her to do good and forbid her from evil, and strive to find a righteous husband for her.