What would happen if someone were to fall asleep with a bad toothache? Will it affect the brain or body?

Possibly. If you can even get to sleep with a bad toothache you should see a dentist. A toothache could be caused by a number of things. The most common causes are an infected pulp that is still vital and an infected pulp which is non vital. The second variety is the one to be concerned about with your question. Bacteria and their toxins are invading the jawbone and blood stream at this point. Get care now.
Pain means infection. If your teeth hurt, you probably have areas of tooth or gum infection. See your dentist asap to get the infection under control. Sleeping isn't an issue, time is. The longer you put off treatment, the harder it will be to control the infection.
Pain. Usually just pain. Infection can spread to other parts of the body, but most often local pain. See DDS ASAP to prevent bigger problems.
Yes both. Having a bad tooth means you are having an infected nerve, a fractured tooth or an abcess...All are bad! all will produce toxins that travel to the rest of your body including your brain..Thje quicker you get this takenm care of the better.