Are some types of anemia inherited?

Yes, several are. Several types of anemia ("low blood", low hemoglobin) are genetic and inherited from one or both parents. Thalassemia is one example, and is quite common. Carriers have mutations on one chromosome and are mild to moderately anemic. Carriers often don't know they are carriers unless a doctor has checked blood tests. A fetus with mutations on both chromosomes may die from severe anemia before birth.
Yes. Yes, some types of anemia are due to inherited or genetic defects. Sickle cell anemia occurs most frequently in people of african, african-american, and mediterranean descent. Thalassemia (or cooley's anemia) is found in people of mediterranean and southeast Asian descent. These types of anemia are extremely common. Another inherited anemia is fanconi anemia, but this is very rare.