An Umbrella, Modular Study Based on Epidermal Growth Factor Receptor (EGFR) Mutation Status

Drug Administration and Study visits for Erlotinib hydrochloride: If you will be taking Erlotinib hydrochloride in this study, you will take Erlotinib hydrochloride by mouth daily at least 1 hour before eating and 2 hours after eating. You will have study visits once a month. If you continue to be on study longer than 2 cycles, study visits may become less frequent. At these visits, the following tests and procedures will be performed: * Your performance status will be recorded. * You will be asked to list any drugs you may be taking, including over-the-counter drugs. * You will be asked about any symptoms you may have. * You will have a physical exam, including measurement of your vital signs. * Blood (about 2 teaspoons) will be collected for routine tests After the first 8 weeks on study, you will have a CT or MRI scan to check the status of the disease. You will have a CT or MRI scan every 8 to 12 weeks after that. You may continue taking Erlotinib hydrochloride for as long as you are benefitting. You will be taken off study if the disease gets worse or intolerable side effects occur. If the disease gets worse, the study doctor may assign you to 1 of 3 studies. The study doctor will discuss this in more detail with you. Length of Study: Your participation on this screening study will be over after all of the screening tests and procedures described above have been completed. This is an investigational study. Up to 102 participants will take part in this study. All will be enrolled at MD Anderson.