Yes. If you are anemic, you should have the underlying cause of the anemia determined and then, if possible, have the anemia corrected. If it is a chronic but stable condition, it should still be followed at regular intervals with periodic assessments to make sure that it is not worsening. Either way, anemia is not a contraindication for surgery.
Yes. Most rotator cuff repairs are done arthroscopically these days, which means minimally invasive and little blood loss. If you are mildly anemic, there should be no problem. If you are severely anemic, then even a small amount of blood loss could be a concern. Severe anemia should be corrected before surgery.
Yes... not bloody surgery. Anemia should not prevent surgery, unless it is severe. In that case, would try to correct the anemia to a safe level to allow for surgery.
Depends on severity. It depends on how anemic you are as well as how your surgeon proposes repairing them. If you have a mild anemia, and you are undergoing an arthroscopic repair, it shouldn't be a big problem. On the other hand, if you have a significant drop in your blood counts (hemoglobin or hematocrit) and you are undergoing a mini-open repair, then the small amount of blood loss may affect you more.