Is the shop considered an unlawful source of livelihood, given the existence of its original owner whose shop was confiscated by the state, and given the actions of the cousins that led to debts and an escalation of problems; and does this entail legal responsibility and debts that must be borne?

What your father did by assuming an unknown debt in exchange for taking the shop is a forbidden error, due to the presence of jahalah (ignorance/uncertainty) and gharar (excessive risk/deception). If the shop belonged to another person and the state seized it without a legitimate religious right, then it is not permissible to acquire ownership of it, and it must be returned to its original owner. Perhaps the cause of the problems that occurred is the seizure of people's rights and properties without right.