No. During the Cold War Western commentators tended to use them interchangeably, both out of force of habit and to play up continuities with the Tsardom, particularly its image as an inherently expansionist empire. For example, George Kennan's famous Long Telegram, having sent the dogs the wrong way with talk of a global ideological struggle, rooted the motivations of the Soviet leadership in a "traditional and instinctive Russian sense of insecurity". Similarly, Churchill rarely, if ever, used the term 'Soviet Union', much preferring 'Russia'. Both implicitly denied the legitimacy or relevance of 1917.
Yet this ignored the small reality that 1917 had happened and that the Soviet Union was not the Russian Empire. Ethnic Russians only comprised approx 50% of the USSR's population and, for all of the complexities in this relationship, the USSR never made use of an 'official nationalism' as the Tsardom had. 'Russia', in English as least, entirely fails to capture this dimension of the Soviet experiment.
(Approaching the question from the opposite angle as Westerners, Solzhenitsyn similarly rejected the twin use of the terms, bristling at the suggestion that Russian culture be equated with Soviet ideology.)
In the Soviet era 'Russia' should refer to the RSFSR. Often this is enough: many studies of 'Soviet Russia' make no pretence at looking to the other national republics and, given the predominance of Moscow, don't need to. But when referring to the polity as a whole then 'Soviet Union' or 'USSR' should be used.
No, but it's understandable that it is. The Russian SFSR was by far the largest of the 15 constituent republics; it was home to a little more than half the USSR's total population, and was the dominant republic both politically and culturally. But the use of 'Russia' as shorthand for 'the USSR' confuses our understanding of what the USSR was: it was a transcontinental superstate, with an enormously complex cultural and ethnic mix. Perhaps the centralisation of power in Moscow and the general dominance of ethnic Russians in the Soviet leadership means that the USSR was distinctly 'Russian' in its external relations — but if you want to understand its society or internal dynamics, you have to take a more nuanced view.
To add to the other fine answers here by u/k1990 and u/International_KB , I would also say no and note it is a bit aggravating to see this Cold War-era terminology used haphazardly because it tends to obscure important facets of the nationalities issue in the USSR.
Although Great Russians made up a disproportionate amount of the senior leadership positions inside the Soviet Union, the Soviet state's relationship with Great Russian ethnic identity, culture, and history was often quite tense and freighted with the legacy of the tsarist empire. Lenin was quite emphatic that the new Soviet state would reject the vestiges of Great Russian chauvinism and actively sought to craft a Soviet family of nations in this new state that would work to create a supranational Soviet identity. Despite such initiatives like Korenizatsiaa (described by the historian Terry Martin as "Soviet affirmative action"), the state gradually prioritized Great Russian culture as the elder brother its family. Yet the Soviets state could not just abandon its commitment supranationalism and often sought to appropriate Great Russian cultural figures as harbingers of the new Soviet order. Soviet pedagogy and other instructional devices like the Great Soviet Encyclopedia took great pains to present men like Tchaikovsky, Tolstoy, or Pushkin as divorced from any problematic class background and argued they were at fundamentally at odds with the tsarist order. In short, Soviet pedagogy emphasized these were people who would have been revolutionaries in 1917, but could not have been because they were born several generations too early. While celebrating these figures in schools and higher culture represented a functional attempt at Russification of Soviet society to an extent, such efforts usually had to have an ideological gloss to be palatable for the Soviet state.
Therein is one of the major problems with using "Russia" interchangeably with Soviet because it erases the tensions implicit in the Soviet approach to its "Great Russian problem." Ethnic nationalism and particularism were vices of capitalism and the Soviet state took great pains to avoid the appearance of extolling Russian ethnicity at the expense of the USSR's nationalities.
