I have a numb tooth and half of my face is swollen. Is this normal?

It can be. If you have swelling, if could lead to a numbing feeling or tooth. The accumulation of fluids and inflammation can produce pressure on tissues and nerves. This pressure will cause loss of feeling in teeth, lips, face, etc. If pressure is too hard or persists too long, it can cause permanent damage and numbness. See your dentist soon.
NO. This is far from normal. Sounds like you could have a severe infection around a tooth. At worst, these kinds of infections are life threatening so see a dentist to take care of this before you end up in an emergency room. It is unlikely that this situation will get better by doing nothing.