The case of Georgia is a particularly remarkable one for the fledgling Soviet Union. In the spring of 1921, the Red Army invaded Georgia and overthrew the Menshevik government, having earlier recognized an independent Georgia in the early summer of 1920; and it is in fact Stalin who advocates for this invasion.

Stalin, even before the Revolution, had established himself as the authority on the “National Question” of Marxism. Stalin (ostensibly with Bukharin and Lenin) wrote Marxism and the National Question, where he charges national identity as irreconcilable with the Marxist project of equality. The Petrograd Soviet - before the Revolution - established the Narkomnats, led by Stalin from 1917 until 1923 (though he functionally exerted less influence into the 1920s), which reflected the state policy of korenizatsiya, where regional languages are considered equivalent to Russian. By the late 1920s we see a dramatic reversal of this policy and the forceful Russification of regional languages.

Throughout 1922, there was a significant conflict on the future direction of Georgia: as a quasi-independent state, as a Soviet Republic preserving its identity, or grouped together with other Caucasian identities, eliding what it means to be Georgian. Stalin and Ordzhonikidze take hardline stances against Georgian nationalism, because for Stalin you are not Georgian but Soviet. By the end of 1922 Lenin himself objected to Stalin's anti-Georgian intrusions, with a politically unsavory disdain to the Georgian cause that returns later, but his failing health left him unable to attend personally to the matter.

The late summer of 1924 sees a dramatic but briefly-lived insurrection in Georgia led by the Mensheviks, squarely rebuffed by the Soviets (and marking the last major rebellion against the early Soviet Union). Many thousands of civilians were executed, and an even greater number of revolutionaries were purged: some tens of thousands of ethnic Georgians were killed by the direction of Stalin as well as the Georgian Ordzhonikidze. Ordzhonikidze would later break with Stalin in the early 1930s, and was likely assassinated in 1937. It's worth centering the role of Lavrentiy Beria - also a Georgian (Mingrelian - thank you /u/kaiser_matias) - in the direction of Georgia in the early 1920s. Stalin’s relationship with Georgia is not entirely one of Russification, and in the 1940s he becomes significantly more sympathetic. Stalin certainly took a markedly oppositional stance in Armenia, in contrast to Lenin (consider the Azerbaijan-Armenian relationship of the 1920s).

To answer the first question, not really. The early Bolsheviks were quite the ethnically diverse crowd (Trotsky was Jewish, Lenin possibly had some Kalmyk or another indigenous Siberian ancestry, Mikoyan was Armenian, etc). The Georgians were also one of the most prominent non-Russian groups actively involved in the early years (Jews and Latvians would be the other major groups), so it was not unusual to have a Georgian working in the party, even in the higher echelons. Georgians in particular had attracted notice within the Bolsheviks as they were some of the more militant and active groups, leading efforts throughout the Caucasus (especially in Baku, where the Russian oil industry was based; a lot of labourers there ready to take in socialist ideas).

As to Stalin's heritage influencing his actions and perceptions, that is something his biographers have been looking at since he first came to power. As mentioned above Georgians took a very active role in leading strikes and demonstrations throughout the Caucasus, and Stalin was very much involved in this, indeed becoming a leader there. He spent quite a bit of time in Baku itself, and before that Batumi (the major port on Georgia's Black Sea coast; where the oil pipeline ended and it was shipped out to the world), and also led efforts in the capital Tiflis (now Tbilisi). And before that he was very much immersed in Georgian life, though he did grow up in an urban setting (not a lot of Georgians did that at the time). He did study at the Tiflis Seminary when he was younger, and his mother had wanted him to become a priest, but while there Stalin lost his interest in religion, especially once he discovered the writings of Marx et al (they were widely circulated in the Seminary, and it is no surprise nearly all the major Georgian socialists studied there at one point).

How much Stalin's Georgianness influenced him is a subject of debate though, and scholars are mixed. Ronald Grigor Suny's recent book on the early life of Stalin makes considerable mention of Stalin's father, Beso, and his perceived "failure" to be a Georgian man (Suny uses the Georgian word "katsi", კაცი, throughout for this feature, which is literally just the word "man"). But he does not go as far as other biographers in saying it was Stalin's rough childhood that led him to be what he became. Instead he suggests that because Beso was not able to perform like a true "katsi" (with all that entails), Stalin was influenced by that, as he later never really adopted a Georgian persona (he never denied his ethnic background, but saw himself as an internationalist).

Likewise he absorbed only part of his mother's religious devotion, and instead of committing to the church he committed to a far different ideology, socialism. With the major role the church played in Georgian national identity at the time (and still does to an extent), it is difficult to say if he would have adopted such fanaticism elsewhere.

Lastly Suny notes that the Georgia of Stalin's youth was a violent place, and he was not immune to seeing that (his father got involved in multiple drunken brawls, Stalin was beaten regularly by both parents, and Stalin even got into some good fights while a student). All this would merge together to make Stalin the person he is, and Suny argues that it was the unique environment of Stalin's Georgia that helped him grow into what he was.

For a comprehensive look at this era of Stalin, I'd highly recommend Ronald Grigor Suny's book Stalin: Passage to Revolution. It was published in October 2020, and I would call it a definitive work on young Stalin. It's also good because he ends with a chapter on the historiography of Stalin biographies, looking at how previous scholars have perceived this period.