Is it considered Islamically forbidden for a sister and her husband to stay overnight in the house of the unmarried sister, which consists of two bedrooms and a living room, especially with the husband using all of the house's facilities? And is fulfilling some needs outside the house with him forbidden?

The sister's husband is a mahram (definition) to her [the questioner], so it is not permissible for her to be in khalwa (definition) (private seclusion) with him, or to sit with him without a mahram, or to go out with him, or to uncover her face to him. It is obligatory for the woman to cover her face from her sister's husband because he is a non-mahram to her. There is no harm in him staying overnight with you if complete caution is observed and there is no fear of falling into what is forbidden. If the house is small, then the sister and her husband should look for other accommodation or reduce the duration of the visit while maintaining complete hijab (definition).