Will a woman be held accountable for her own money or her husband's? And does he have a documented right to it because he immigrated with her and left his original job? And is her current situation Islamically sound, or is it better for her to stay at home?

The owner of wealth is responsible for its source and its expenditure. If the wife owns money and authorizes her husband to manage it, she is responsible for it, and he is accountable for his trustworthiness and for assisting her in its expenditure. The default is that what a wife earns from her job is her private property unless the husband explicitly stipulates otherwise. And what the husband does in managing her affairs is a voluntary act of kindness unless they agree on compensation.

If the wife willingly gives her husband something or registers property in both their names, it is an act of charity and kindness. If the wife's work is free from Sharia prohibitions, does not affect her worship or family duties, and provides money that helps her family, then continuing in it is good. However, if it entails Sharia prohibitions and the husband is capable of providing for the family, then it is preferable to leave it.