I showed my design to a friend, and then he copied it and submitted it for his team. Now he’s ahead of me in the standings–with the design he took from me! Can you disqualify him for cheating?

No, sorry. What your (former) friend did was not nice, but it was not cheating. There are no restrictions on collaboration during the Qualifying Round. And the rules state, each team is responsible for safeguarding its own designs. When you shared your design with your friend, you gave up any right to claim it as your own.