This phase II trial studies the best approach to combine chemotherapy and radiation therapy (RT) based on the patients response to induction chemotherapy in patients with non-germinomatous germ cell tumors (NGGCT) that have not spread to other parts of the brain or body (localized). This study has 2 goals: 1) optimizing radiation for patients who respond well to induction chemotherapy to diminish spinal cord relapses, 2) utilizing higher dose chemotherapy followed by conventional RT in patients who did not respond to induction chemotherapy. Chemotherapy drugs, such as carboplatin, etoposide, ifosfamide, and thiotepa, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Radiation therapy uses high energy x-rays or high-energy protons to kill tumor cells and shrink tumors. Studies have shown that patients with newly-diagnosed localized NGGCT, whose disease responds well to chemotherapy before receiving radiation therapy, are more likely to be free of the disease for a longer time than are patients for whom the chemotherapy does not efficiently eliminate or reduce the size of the tumor. The purpose of this study is to see how well the tumors respond to induction chemotherapy to decide what treatment to give next. Some patients will be given RT to the spine and a portion of the brain. Others will be given high dose chemotherapy and a stem cell transplant before RT to the whole brain and spine. Giving treatment based on the response to induction chemotherapy may lower the side effects of radiation in some patients and adjust the therapy to a more efficient one for other patients with localized NGGCT.