Essentially, there is not much difference between an injury or a wound. Injury might be slightly more formal. Another difference is in the connotation. If you said you received an injury, I am likely to think you did it playing sports, or being active (running, walking). Also I would think that you sprained an ankle or broke a bone or pulled your back. I would be less likely to think that you received a scratch that opened your skin, causing you to bleed. If you said you received a wound, I am more likely to think that you are/were bleeding as a result of it. We say an open or "gaping wound" for a wound that requires a bandage to prevent bleeding or infection. Wounded also sometimes carries a connotation of battlefield injury. It is not wrong to say someone was injured on the battlefield, but in war, they count the number of dead (KIA) and wounded.
As far as your second question, we generally say "it pains me" when we are feeling emotional pain. For example, "it pains me to see you looking so sad." The phrase "hurts me" can be used both for emotional pain and for physical pain.
Hope that helps!