What is the ruling on playing cards once a week if it is free from wasting time, delaying prayers, distraction from acts of worship, and does not lead to animosity or problems between players? Is there a fixed Sharia ruling for this game, and what is its evidence?

There is no harm in playing cards if the Sharia (definition)-ordained prohibitions are avoided therein, and that the play is not for a prize, lest it become prohibited gambling. The basic principle regarding things is permissibility unless there is evidence to the contrary.