The existence of a negative externality regarding the production of a good results in the underproduction (and, thus, underconsumption) of a good at a market equilibrium. True or False?

This is false. Firms for the most part do not factor in negative externalities when producing their goods. If the negative externality is foreseen as also being a burden on their costs, then the firm may act accordingly to not produce as much of the good or shift production so to minimize the externality. Firms will continue to produce their good as long as the party suffering from the externality does not act. For example, if a trucking company produces a negative externality for a neighborhood because of all the noise, the company is not going to stop production unless there is an ordinance or legitimate complaint filed. Usually the local government would intervene.