Are half-siblings entitled to claim inheritance of an apartment that the family obtained as displaced persons, even though they did not participate in the search for family shelter nor provide any assistance?

If the state gifted the apartment to your father as property, then it belongs to all his heirs, including the paternal half-siblings, because they inherit by lineage. If the gift was to all of you, then only the father's share is inherited, and his paternal half-siblings would have a share in it.