I went to the blood bank and they told me I was anemic; how could I have iron overload at the same time?

Hemolytic anemia. Individuals with hemolytic anemia, an anemia due to excessive red cell destruction, typically have normal or higher iron level. Therefore, ingestion of iron supplement or receiving blood transfusion will result in iron overload.
Possibly. It depends on the cause of the anemia. When the body senses anemia, absorption of iron from the GI tract increases. That's good if the cause of the anemia is iron deficiency, but not helpful with other causes. Longstanding anemia not due to iron deficiency could cause iron loading. Inherited anemias, especially thalassemia, can cause iron loading.