Philosophy was an extremely touchy discipline in the USSR because the state ideology was explicitly founded on its philosophical rightness. In other words, it was inherently politicized, and in the first decades of the USSR's existence, one of the most highly politicized areas of study. Soviet philosophers not only fought amongst each other (sometimes denouncing the philosophy of others as bourgeois and anti-Soviet, which could have disastrous personal and professional consequences depending on the time period), but also amongst other disciplines (Soviet philosophers attacked Einsteinian physics, for example, as bourgeois and anti-materialist, requiring vigorous defense by physicists like Vladimir Fok).

So while it would not have been limited to Marxist theory per se (as written and interpreted by Marx, Engels, Lenin, and Stalin), there were several very specific tenets that ran through it and could not be contradicted, even implicitly.

The result is that in many respects it was incredibly conservative. When contradiction of dogma can lead to your being sent to a labor camp, you work hard to show that you can never been caught contradicting. The consequence here is that from a Western perspective, most of what came out was ideological dreck, similar to Soviet political science, sociology, and any other discipline whose answers were already dictated by ideology. Not all academic disciplines were so hidebound; the physicists eventually managed to gain a lot of autonomy (in part because of their military usefulness) and Soviet physics was considered world-class. Soviet mathematics managed to avoid politicization for the most part, as well. Biology notably did not (Lysenkoism, etc.).

This is not meant to be a simple argument about the role of politics in knowledge, or to make it seem like the Soviet Union was the only place that politicized its academic work. But Soviet philosophy was particularly hidebound.

As for whether Western philosophers were considered suspect — yes, definitely. I don't know if the Soviets gave Heidegger any notice, but the fact that he was a German active at the time of the Nazis would have made him someone you would not want to cite. (This is separate from the more complicated question of Heidegger's connections to Nazism.) There are examples of some Soviet philosophers being interested in Western writers like Spengler and Bergson, but having immediately to fend off attacks of being bourgeois pessimists and things of that nature.

Anyway, the above is a sketch of an answer — I know much more about the cases of physics and biology than philosophy, though the philosophers were active in both of those cases as well (again, they were among the more doctrinaire of the disciplines). For a general overview, Loren Graham's Science in Russia and the Soviet Union: A Short History is a classic, and has chapters on the role of dialectical materialism in particular in Soviet philosophy and science. (Dialectical materialism was the official state-sanctioned philosophy of science.)

Most of the major philosophical figures of classical philosophy would have been available in translation. Soviet censors during and after the Thaw tended to be preoccupied with contemporary works, so classic literature in the Western canon such as Dickens generally got a pass, and this included classical philosophy. The translations were under the auspices of state institutions, frequently in large multi-volume sets, and featured commentary by various Soviet specialists to frame each work.

Philosophy as a discipline occupied an anomalous position within the Soviet academic hierarchy. The Soviet academic establishment simultaneously decreed philosophy to be a vital component of the new Soviet man's education, but approached the teaching of it in a highly constricted fashion. Just as in the West, most Soviets pupils encountered philosophy through advanced education. A required component of most Soviet PhDs and other graduate programs was to study philosophy. Here, the Soviet pedagogical approach to philosophy splintered along several lines. Some teachers would emphasize each philosopher as reflective of their respective socioeconomic epoch. Another common approach was to teach the history of philosophy as a teleological process leading to the discipline's culmination into Marxist-Leninism. Still other teachers would reduce philosophical debates into a struggle between idealists and materialists.

The result of this pedagogy was that while some Soviet philosophers were aware of Western developments in philosophy like the Frankfurt school, discussion of these figures and their works seldom assumed a prominent place in Soviet philosophical discussion. In the conservative Brezhnev era, the Soviet philosophical establishment would critique the Frankfurt school as a deviation from standard Marxist-Leninism. Soviet journals of this era would critique figures like Lukacs as "pseudo Marxists" and if they invoked their philosophical tenets at all, it frequently was in a point by point refutation of their deviations from the Marxist-Leninist paradigm. By the Glasnost era, both access and knowledge of the various critical theorists was much better known in Soviet philosophy. There was much greater contact between Soviet philosophy departments and their Western colleagues and Soviet journals began to invoke postwar Western philosophers to discuss their ideas, but not to rebuke them. Despite the new openness of the 1980s, Soviet philosophers generally did not accept the tenets of philosophers like Adorno and largely considered them to have been an intellectual cul de sac.

Sources

Ballestrem, Karl G. "Soviet historiography of philosophy." Studies in Soviet Thought 3, no. 2 (1963): 107-120.

Blakeley, Thomas J. "Lukács and the Frankfurt School in the Soviet Union." Studies in East European Thought 31, no. 1 (1986): 47-51.

Neumaier, John J. "The Frankfurt School in Soviet Eyes." Philosophical Forum 17, no. 4 (Summer1986): 322-334.

Zweerde, Evert. "Recent developments in Soviet historiography of philosophy." Studies in East European Thought 39, no. 1 (1990): 1-53.

_. Soviet historiography of philosophy: istoriko-filosofskaja nauka. Vol. 57. Springer Science & Business Media, 1997.