Is it permissible to pay the hospital bill from the wife's private money or to apply for a job to pay it without the husband's knowledge, given his refusal to pay it, and knowing that the bill is considered a debt upon her?

It is not permissible to use a credit card if it incurs usurious interest, except in a case of extreme necessity, such as to avoid imprisonment. Otherwise, in the absence of necessity, the lender must grant the insolvent person respite. If the card is permissible and you are compelled to use it to avoid imprisonment, then it is permissible to use it whether the husband knows or not. And if the company calculates usurious interest on the bill without your direct involvement in usury, then there is no sin upon you. It is not permissible for a woman to work outside her home without her husband's permission, and it is not permissible for her to work in a mixed environment except out of necessity.