These are good and important questions, but in order to answer it properly, I'll have to disentangle them a bit, because there was a rich and diverse Judaism within both the Soviet Union and tsarist empire that works against easy answers.

The tsarist state considered the Jews that resided within its empire as subjects to the crown, and increasingly fell under the category of inorodtsy in the nineteenth century (inorodtsy essentially means subject aliens). The majority of the empire's Jews lived in a region called the Pale of Settlement, although there were Jewish communities within the Baltic and Russian areas. Most Jews could not migrate outside of the Pale unless they had a profession or other factor in their favor such as military service (unlike other inordtsy groups, Jews were subject to conscription in the nineteenth century). Thus the Jews outside of the Pale (Lithuanian Jews being a notable exception- it's complicated) tended to acculturate to Russian mores. As the nineteenth century wore on, political activism among Jews within the Pale tended to drift towards either Zionism or Bundism. The former was a form of Jewish nationalism that sought to revive Hebrew and held Jewish culture as distinct from neighboring ethnic groups. Bundists also hewed to Jewish particularism, but within a Marxist-socialist context and they found a receptive audience in the Pale's major cities like Lodz, which were rapidly industrializing and had a strong Jewish population. Although traditionalist Jews within the Pale were a strong element of society, both Zionism and the Bundists received a boost in membership from the pogroms of the late nineteenth century. This anti-Jewish violence fostered counter-organization as the tsarist state was unwilling or unable to protect the lives of its Jewish subjects. The First World War and the dislocation of the Civil War led to even more pogroms and antisemitism flourished among White forces.

Although Zionists made some tactical alliances with revolutionary forces in the changed situation of 1917, the Bundists found it much easier to ally with the Reds. A number of Jews also flocked towards the Bolsheviks for a number of reasons. The Red Army in many ways put a stop to the pogroms within Ukraine and the Bolsheviks eliminated the tsarist legal restrictions on Jews' movement and occupation. As the Soviet Union emerged from civil war, it found it needed a trained and educated bureaucratic civil service and many Jews, often Bundists, staffed these lower- and mid-level positions since Jews by and large had higher literacy rates than other members of the population. The Soviet state also encouraged Jewish cultural projects and held that all its citizens were equal before the law irrespective of ethnicity or religion.

It is tempting to view the 1920s through an idealized lens for Soviet-Jewish relations, however such idealizations are deceptive. The crux of the "Jewish Question" was over whether or not being Jewish was an ethnicity or form a religious practice. If one ascribed to the idea that Jewish identity was ethnicity, then Jews should be welcomed into the brotherhood of Soviet nations; conversely, if Judaism was a religious practice, then it was an archaic form of identity that could never square with the new Soviet man the USSR was creating. The fractious nature of this debate was apparent in the special Jewish section of the Communist Party (evsektsii) which unlike other special ethnic sections was explicitly temporary. By the late 1920s, the Stalin turn saw a partial victory of the perspective that Jewish was a religious practice and many of these special privileges (Terry Martin terms this Soviet-style Affirmative Action) granted to Jews were repealed. Stalin also arguably appealed to coded antisemitism in deposing of his rivals like Trotsky, who had a taint of a Jewish background.

Although Soviet Jews found the 1920s privileges curtailed under Stalin, the state did not openly hew towards an antisemitic line. Open statements of antisemitism were disciplinary offenses if reported to the state's authorities. The rise of the Third Reich also precluded a open antisemitism as German racism became lumped together as an ancillary component of Nazism's anticommunism within Soviet propaganda. Majority Jewish areas like Minsk openly embraced their identity as Soviet Jews as the ideological war between communism and fascism sharpened during the late 1930s. Despite this early form of Soviet patriotism, many Jews became victims of the Terror.

The Great Purges of the late 1930s had a large number of Jewish victims. Some of this was due to a covert antisemitism amongst security officials, but keep in mind many of the purgers were themselves Jewish in origin. Part of the reason for the number of Jewish victims is a function of how the Purges were conducted. If one had a questionable ideological background pre-1917, the odds of surviving the Purges lessened a great deal. Since many Jews involved in the Soviet state were former Bundists, they had a black mark on their biographical commitment to revolution that they could not erase like their gentile colleagues. Furthermore, informal antisemitism also meant that many non-Jews within the state saw Jews as outsiders, which was dangerous in the context of the circular firing squad of the Purges. Many of the mid-level managers when asked to name names by security forces would sacrifice outsiders before they would give up their inner circle.

The Second World War also saw an uptick in Russian antisemitism as the state no longer bothered to discipline antisemitic outbursts among the public. The life or death struggle with the Third Reich also precluded any acknowledgement by the USSR that antisemitism existed within its own borders because that would mean the Soviet Union shared common ground with its existential foe. A rekindling of antisemitism also received further sanction from the upper echelons of Soviet society as Stalin's paranoia directed itself towards Jewish elements within the USSR. Despite this postwar upswing, antisemitism within the USSR shared little with the antisemitism of Nazi Germany; Jews were still Soviet citizens and if they acculturated, they faced little formal discrimination by the state. That being said, Jews were an ethnicity that the postwar Soviet state possessed little love for, and the state was indifferent to complaints about antisemitism among the general Soviet public. The Cold War and the Arab-Israeli conflict added a further strain as the Soviets increasingly backed the Arabs in their public diplomacy. This meant that Zionism became linked in Soviet discourse with Western imperialism, so then to emphasize being a Soviet Jew during the postwar meant identifying with an ideological opponent of the state, unlike the 1930s. Thus to be a Jew in the postwar USSR could be characterized as a begrudgingly accepted outsider.

Sources

Abramson, Henry. A Prayer for the Government: Ukrainians and Jews in Revolutionary Times, 1917-1920. Cambridge, Mass: Distributed by Harvard University Press for the Harvard Ukrainian Research Institute and Center for Jewish Studies, Harvard University, 1999.

Bemporad, Elissa. Becoming Soviet Jews: the Bolshevik experiment in Minsk. Bloomington: Indiana University Press, 2013.

Gitelman, Zvi. A Century of Ambivalence: The Jews of Russia and the Soviet Union, 1881 to the Present. Bloomington: Indiana University Press, 2001.

Nathans, Benjamin. Beyond the Pale The Jewish Encounter with Late Imperial Russia. Berkeley: University of California Press, 2002.

Slocum, John W. "Who, and when, were the inorodtsy? The evolution of the category of “aliens” in imperial Russia." The Russian Review 57, no. 2 (1998): 173-190.