Is the money the questioner received from the state, after claiming to be searching for work, while knowing she was lying, unlawful (haram) or lawful (halal)? And what should she do with it? And is it her right or her husband's right?

As long as the state provides a salary to those who seek employment and not to those who stay at home without seeking employment, it is impermissible to take the salary if there is no job search, because observing the condition of the giver is obligatory. What has been taken from these salaries is not permissible, and one must repent to Allah for the lying and deception, and return what was unlawfully received to the state by any means without requiring its notification. If the money cannot be returned immediately, it becomes a debt that must be repaid whenever possible.