Is speaking with the husband's sister about worries – without mentioning the husband's private habits – considered a breach of the oath requested by the husband, who suffers from schizophrenia, not to reveal what goes on in the house?

If the husband has made his wife swear an oath without the oath being tied to a right, then the reference for the oath is the wife's intention, not the husband's. If she intended not to speak about the husband's private habits only, then she has not broken her oath. However, if she intended not to speak about anything concerning the household affairs, then her conversation with his sister would be considered a breach of the oath. If the one making the oath is not unjust, and the one for whom the oath is made is not wronged, then the oath is based on the intention of the one swearing it.

Regarding the wife speaking about her husband, if there is a benefit in it, then there is no harm. Otherwise, it serves no purpose and may lead to increased suspicion and problems. The wife might also fall into backbiting, which is forbidden, based on the Prophet's (peace be upon him) saying: "Your mentioning about your brother what he dislikes."