Yes, the learning of Russian in the German Democratic Republic's schools was obligatory, at least for the majority of the history of the GDR.

Foreign-language education was an important goal of East Germany's socialist education policy. The 1959 Law on the Socialist Development of the School System in the German Democratic Republic (link in German, my translation) declared that "[t]he imparting of foreign languages, especially the Russian language, serves the consolidation of friendship between nations, and enables the pupils to familiarize themselves with the progressive achievements of other peoples." The 1965 Law on the Unified Socialist Educational System (link in German, my translation) similarly stated that "[f]oreign-language instruction is to promote understanding between nations."

While special emphasis was placed on Russian, foreign-language education was by no means intended solely to ideologically indoctrinate pupils or students. It had a utilitarian function, as well. The 1965 law again: "Education in Russian and in a second foreign language [at the University level] shall enable the students to follow the foreign-language literature in their field of specialization and to communicate regarding specialist questions in the foreign language." (For pre-university pupils, the goals were less ambitious, but no less practical: "Pupils are to be enabled to communicate in the foreign languages and to understand simple texts with general as well as popular-science contents.")

As of the 1965 educational law, Russian language education was made mandatory for pupils beginning in the so-called Mittelstufe - roughly equivalent to the American middle school - which occured before children were segregated on the basis of their future educational or professional paths. Pupils were also required to study a second language, which the educational law envisioned would be - but did not require to be - English. According to the May 1989 publication "40 Years of the German Democratic Republic," (link in German) distributed by the GDR's Central State Administration for Statistics, fully 11% of the of the instruction given to pupils in the polytechnical secondary school - of which the Mittelstufe was a part - was devoted to foreign-language education.

Whether and how this extensive education in the Russian language affected the culture of Eastern Germany is hard to say. Even before the end of the Cold War, many scholars of Germanistik noted the divergent development of the German language in the GDR, particularly in terms of vocabulary. There were words used in the East that were not used in the West, and vice versa (e.g., Kaufhalle in the East and Supermarkt in the West). Before 1989 there was a cottage industry of "GDR Dictionaries" in West Germany. There was also arguably a grammatical dimension to this divergence. But the differences could be overstated: A lot of the peculiarities of "GDR German" were in fact just Amtssprache - byzantine formulations used only in government affairs, not touching the everyday language of average German people. It's also hard to say how much of any of this was due to the influence of Russian-language education. You could just as easily argue that any divergence between eastern and western Germany was the consequence of the latter's prolonged exposure to American and British language and culture.