Do people with cancer always become anemic?

Almost always: A few rare caners, like renal cell carcinoma and vascular tumors of the brain may cause polycythemia, i.e., increased red cells. Most turmor will cause anemia, even if only in the terminal stages.

Many causes: Cancer patients can get anemia from blood loss, chemotherapy and the cancer itself- so eventually, lot of patients have anemia (also depends on stage/extent of treatment).