Is it permissible to file a lawsuit against a foreign city to claim financial compensation for the death of a daughter in a swimming pool that lacked safety instructions, and is it permissible to spend part of this money on building a mosque and an Islamic school and to benefit from the rest?

If the city administration was negligent in the commonly accepted safety and security measures for pools, your friend is permitted to file a lawsuit demanding the blood money (diyya) for his daughter. He may dispose of this money as he wishes within lawful bounds, and dedicating a portion of it to charitable deeds as an act of devotion for her is permissible. However, if the administration was not negligent, it bears no responsibility, and it is not permissible to file a lawsuit against it.