Half of my right hand feels strange, and kind of hurts a little, what is wrong with it?

Hard to say: Two common things to think about are nerve issues. 1) carpal tunnel syndrome causes numbness, pins and needles or sometime pain in the palm, thumb, index and middle fingers. 2) ulnar nerve "issues" that cause similar symptoms but in the middle, 4th and pinky fingers and the back of the hand. These are just some of the possibilities and you are best suited by seeing your doctor to review the issue

Hand pain: It's impossible to say based on just what you said. You could give a lot more details and resubmit. Otherwise, it really could be something important or could be something that will go away on its own. Consider seeing a doctor for evaluation if the symptoms are significant or if they get worse or don't go away.