When Kara pulls Mister Mxyzptlk into her apartment, she's wearing her glasses. The next shot as Alex closes the door shows her without her glasses. There's a cut to Mister Mxyzptlk's face, and when it cuts back to Kara, she has her glasses on.
During the courtroom scene, Lena "pleads the fifth" when asked to reveal Supergirls identity, which media often misinterprets. The fifth amendment states, "No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation." Basically unless, revealing Supergirls identity would implicate Lena's involvement in a crime, pleading the fifth would not apply. She could only refuse to answer the question and possibly be held in contempt.