From pre-history to the advent of antibiotics and dental care, what were a persons options with rotting or infected teeth? How common was death caused by an infected tooth?
I can see simple removal, but what about cases where there was nothing left but the root and pieces of tooth embedded in the gum? Did it eventually just fall out, and the hole seal up if the person managed to avoid infection?
Mostly they just suffered through. In some cases they had teeth pulled. In general, prior to modern highly starchy highly sugary diets people didn't have quite so many teeth problems, though even so any dental problems that remain without remedy tend to be extremely painful and life altering.
There are many historical stories of people laboring through tooth and mouth pain. The oldest historical accounts of tooth problems are seven millenia old from Sumeria. Historically there was a misconception that cavities and tooth decay were caused by a "tooth worm", which is a surprisingly common belief across many different separate cultures. This might be due to the appearance of the tooth nerve, which is somewhat worm-like.
As an example, Queen Elizabeth I of England had a history of bad teeth, and suffered from toothaches from time to time. In one case she had such a severe toothache that she couldn't sleep or get much done and eventually had the tooth extracted.
Historical records are not really good enough to get reliable statistics on dental issues as a cause of death, though there have been studies on remains to estimate such statistics (such as this one) which generally come to the conclusion that poor dental health contributed to higher mortality.
Here's a quick and dirty summary from a document from the NHS (in the UK) about dentistry history:
Sumerian texts around 5000 BC is the earliest we have for sources about dental decay. They believed they were caused by some sort of tooth worm.
Throughout the ages a number of things were attempted to stabilize ailing teeth, including using metal wires to secure down teeth that are loose or broken jaws. Ancient Greek writers such as Hippocrates and Aristotle both wrote about these methods.
But primarily, I am responding because I have some knowledge of Chinese practices during the Tang dynasty. There's quite a lot of primary source literature about medical practices in Chinese medicine, and amalgams (fillings) were used as early as the Tang dynasty. We have a complete text about medicine from a doctor named Su Gong (or Su, Gong in the Western world) from this period. I've read some interesting secondary sources about medicine use in the royal family (in context of biographies of said families), and it would include things like painkilling herbs in a paste applied to the teeth, but they were all written in modern times so I'm not sure how much primary support there is for that.
Sources: Czarnetzki, A.; Ehrhardt S. Re-dating the Chinese amalgam-filling of teeth in Europe. http://completedentalguide.co.uk/history-of-dentistry/