After completing secondary school in Vienna's 6th district, Friedrich Trauth studied Greek and Latin and then geology and palaeontology, took further natural science courses and obtained his doctorate in 1908 sub auspiciis imperatoris. While still a student, he co-founded the Geologische Gesellschaft (Geological Society) in Vienna in 1907 and remained active in it for many years in various functions. Even before graduating he started work in early 1908 in the Geological-Palaeontological Department of the k. k. Naturhistorisches Hofmuseum (Imperial Royal Natural History Museum) in Vienna, initially as a volunteer, then as an assistant in 1912, a curator 2nd class in 1921, and later a curator 1st class. He worked in particular in the administration and carrying out an inventory of the Geological-Palaeontological Department collections, identified many objects, rectified incorrect identifications by earlier researchers, and labelled thousands of objects. In 1926 he habilitated at the Technische Hochschule in Vienna in geology and palaeontology. He went on to teach there and was awarded an extraordinary professorship in 1935, becoming director of the Geological-Palaeontological Department in June 1936. From February 1941 to April 1945 he was the deputy of Hans Kummerlöwe, director of the Naturhistorisches Museum (NHM) (Natural History Museum), and was responsible in this function for salvaging the NHM's collections. After the war he organized the return of the stored items.

As a student, Trauth had joined the Deutscher Schutzverband Südmark and Deutscher Schulverein. From 1933 to 1938 as a public servant he was a member of the Vaterländische Front (Fatherland Front). According to a political assessment during the Nazi period he had been "indifferent" before the annexation of Austria to the Nazi German Reich. He states that he applied in June 1938 for membership of the NSDAP and joined the Nationalsozialistische Volkswohlfahrt (NSV) (National Socialist People's Welfare) and the Reichsluftschutzbund (Reich Air Raid Protection League) that same year. In July 1945 he had to register as a former National Socialist. He claimed never to have been a prospective member (Parteianwärter) and in November 1945 applied to the municipal department in Vienna's 7th district for deletion of his name, which had been entered in the register of National Socialists "by mistake". In December 1945 Senate No. 8 of the Sonderkommission I. Instanz at the Staatsamt für Volksaufklärung, für Unterricht und Erziehung und für Kultusangelegenheiten (State Office for Popular Enlightenment, Schools and Education, and Cultural Affairs) judged that Trauth could continue to work at the NHM but not in an executive function. The Ministry of Education followed this recommendation in early 1946. During the denazification trial, Trauth attempted to prove his supposed rejection of Nazism by citing his continued correspondence after the annexation of Austria to the Nazi German Reich with Georg Rosenberg (1897–1969) and Martin Glaessner (1906–1989), who had long-standing contacts with the NHM and had been persecuted by the Nazis as Jews. Following Trauth's application, the Complaints Commission in the Ministry of the Interior decided in November 1947 to remove Trauth's name from the list of registered National Socialists. Trauth was reappointed department head. He retired officially in 1948 but remained in the department until mid‑1949. He was made a Hofrat in 1949 and honorary member of the Geologische Gesellschaft in Vienna in 1954. Finally, in 1958, the University of Vienna awarded him the Goldenes Doktordiplom (golden doctor's degree).