Even though you didn't underline anything, there is only one pronoun in this sentence - the word them.
The case of this pronoun is objective - it functions as an object in the sentence. It is really easy to determine this - all you have to do is ask the questions whom or what. Whom did the designer dress in black and teal costumes? And the answer is - Them.
This way you know that the pronoun is also the object; to be more specific, it is a direct object.
Therefore, the correct answers are case: objective; function: direct object.
