There are two basic forms of thalassemia, depending on which of the hemoglobin genes are affected (hemoglobin is made from both alpha and beta globin). These are alpha thalassemia and beta thalassemia. Technically, Cooley's anemia refers to beta thal, but the focus of CAF is on all thalassemias.
Sufficient hemoglobin levels are somewhat subjective and have to be partly determined by the health of the patient. With HbH, you may see normal Hb levels in the 8-11 range, and as long as this is maintained, transfusions are generally not required. And yes, you should write down and ask every single question on your mind when you talk to the hematologist.