Is it permissible to make mandatory payments to a family fund to cover future blood money (diyat) or compensations resulting from the actions of its members, knowing that whoever does not pay will bear their problems alone?

There is no objection to establishing charitable funds if the contributors are volunteers and willing, and not coerced, as this falls under cooperation in righteousness and piety. There is no issue in preventing those who did not participate in the fund from benefiting from it. As for the blood money (diyya) borne by the 'aqila (the male relatives of the offender on his father's side), they are fundamentally obligated by Islamic law to pay it. If anyone refuses to pay, the killer by mistake may resort to the court.