Is the financial compensation received from a car insurance company for an accident that resulted in severe physical injuries considered lawful (halal) or unlawful (haram), especially since insurance is mandatory and the compensation does not cover the full extent of the physical and psychological damage and suffering?

It is permissible for you to take compensation from the insurance company if it does not exceed the actual damage incurred by you. As for the agreement with the lawyer, there are two scenarios: The first is to agree with him on a known amount if he obtains the compensation (ju'alah - a reward for a service), and the reward must be known. This is permissible if the reward is the difference between the expected amount and the amount paid to you. The second is to agree with him on a known work for a known wage (ijarah - a lease or hire agreement), and this is permissible provided the work is clearly defined to remove ambiguity. However, for the client to waive the case to the lawyer in exchange for a certain amount, and then whatever additional amount the lawyer obtains is for him, this is not permissible due to ambiguity (jahalah) and uncertainty (gharar).