How would one categorize a conflict involving two people engaged in name-calling and personal attacks on each other?
Interaction between two or more individuals involving name-calling and personal attacks, besides being immature, could be categorized as "incivil," "confrontational," "argumentative," or "aggressive." Probably the most appropriate of those labels, though, is "incivil." Incivility is defined as "the quality or state of being uncivil," and commonly refers to the use of name-calling and personal attacks in the course of communications between two or more people. [Definition is from Merriam-Webster.] Depending upon the degree of incivility and aggression, this behavior may be legal assault as defined by Tort and Criminal law:
Intentionally putting another person in reasonable apprehension of an imminent harmful or offensive contact. No intent to cause physical injury needs to exist, and no physical injury needs to result. (LII, Cornell University)
A 2011 article in U.S.A. Today described a study by the American Psychological Association that concluded that workplace incivility was on he rise in the United States, a situation exacerbated by the economic difficulties and high unemployment that have characterized American society for the past half-dozen years. According to the article, the APA study found that "'75% to 80# of people have experienced incivility. It's a growing and prevalent problem,' said Jeannie Trudel of Indiana Wesleyan University-Marion'." ["Incivility a Growing Problem at Work, Psychologists Say," U.S.A. Today, August 9, 2011]
In short, the conduct described in the question can best be categorized as incivil or, depending on excess and degree, assault.