Are funds awarded by the court halal or haram? And is accepting the driver's offer to drop the case in exchange for a larger sum halal or haram?

The amount determined by the court as compensation for the harm caused by the driver is known as "governmental compensation" (hukumat al-adl). It is an obligatory payment estimated in criminal cases where there is no specific legal valuation, and there is no objection to accepting it. Your relinquishment of the case in exchange for this amount is permissible because it is a waiver of a personal right, as the Prophet, peace and blessings be upon him, said: "Reconciliation is permissible among Muslims, except for a reconciliation that makes a lawful thing unlawful or an unlawful thing lawful." As for the imprisonment of the driver, it is a matter for the court and not for you, because it is a discretionary punishment (ta'zir) aimed at protecting society and deterring drivers. It is considered a right of Allah and of society in general, so you do not have the right to waive it or take compensation for it. Rather, the matter is left to the court to punish him or pardon him.