Is it permissible for her to remove her hijab to work in France, or to forge work documents to obtain residency for her husband?

The hijab is a religious obligation, and it is not permissible for a Muslim woman to remove it except out of necessity. There is no necessity in what you have mentioned. Lying and deception are known prohibitions in the religion of Allah Almighty. Therefore, it is not permissible to undertake such actions, nor does this person obtaining residency in that country make it permissible.