In Book 13 of The Odyssey, why does Odysseus test Athena?
Only one question allowed on enotes I am afraid! You need to make sure that you do not ask multiple questions. I have edited your question accordingly to focus on the first element of your question.
I assume that you are referring to the meeting that Athena has with Odysseus when she is disguised as a young boy and reveals to Odysseus that he is on Ithaca. Strictly speaking, Odysseus doesn't know that he is conversing with his patron goddess, and so he doesn't try to trick her directly. However, his desire to tell a string of lies is characteristic of the kind of person that we have seen Odysseus to be throughout the book up until this point. He will always deceive and disguise himself. He delights in using his brain rather than triumphing through his strength, and Athena himself rather endearingly says that this is something of a habit for Odysseus. Of course, these are precisely the same tendencies that she herself displays, making their relationship uniquely appropriate.