Is it permissible to demand financial compensation for an injury sustained by one of two brothers in a fight that occurred when he attempted to take iron from a house, after the police intervened and a lawsuit was filed against the assailants?

It is permissible, Islamically, to file a lawsuit and claim compensation for the damage inflicted upon the hand. If the damage warrants qisas (retaliation in kind), the victim has the right to exact it from all involved parties if they participated. If it does not entail qisas, then arsh (compensation for injury less than death) or diyah (blood money) shall be borne by all of them collectively. Furthermore, it is permissible to demand compensation for any material damages you incurred as a result of the assault, and the lawsuit should be filed to deter the aggressors.