Is the transaction in which the father mortgaged the car for 100,0, with monthly payments of 5,0, considered usury (riba)? And do the father's money and his new salary become unlawful (haram)? What should the children do regarding this, and is their taking from the father's salary considered unlawful (haram)?

The aforementioned transaction is unequivocally usurious, because in reality, your father pays 5000 monthly for the debt he owes. The Quran invalidated this type of usury in His saying, the Almighty: "O you who have believed, fear Allah and give up what remains [due to you] of interest, if you should be believers. And if you do not, then be informed of a war [against you] from Allah and His Messenger. But if you repent, you may have your principal – [thus] you do not wrong, nor are you wronged."

As for the mortgage in it, it is a stratagem to cover up the usury that is occurring. However, your father's money does not become unlawful for you and your siblings, and this does not affect the permissibility of your benefiting from it; for the unlawful pertains to his liability, not to the money itself.

Regarding his work in the company, if it is in a permissible field, then it is lawful; otherwise, it is unlawful, and the salary may consequently become unlawful.

If your father's money is mixed between unlawful and lawful, then it is permissible for you to benefit from it absolutely. But if it is all unlawful, then it is not permissible except if you are in dire need of it, like the need of a poor person. In that case, you may take from it according to your need, because if unlawful money is given to a poor person, it does not become unlawful for them but rather lawful and good.