This is a topic I have long been looking at, and have written and researched about for some time now. And the short answer would be, initially yes, there was special treatment for non-Russians, but like most answers, it's complicated.

Nationalism and communism did not work very well. It was felt that nationalism was a distraction meant to keep the proletariat fighting each other instead of the bourgeois, and thus it was something that had to be moved past. However Lenin felt that it was important for "undeveloped nations" to reach the stage of nationalism, as it was seen as a major stepping stone towards socialism and thus communism. Thus paradoxically the early Bolshevik policy favoured developing national identities, though crucially this was only for those so-called undeveloped nations (essentially anyone who wasn't Russian and European/Western; Armenians and Georgians were grandfathered in as well as they had a centuries-old literary tradition, which I'll touch on below).

This also meant that the Bolsheviks had to determine who existed within the Soviet Union, meaning they would have to classify everyone into a national identity and then they would be able to see who needed help. Over 100 different national groups were eventually delineated, though over the years their numbers and groups shifted based on a variety of ethnographical and political factors.

Anyways, most of these groups (like the Kalmyks, the Buriats, the Yakut, the Abkhaz, Turkmen, and so on) were considered primitive and backwards. Many did not have any writing system, much less a concept of what it meant to be an ethnic Turkmen or something. So efforts were made to help create writing systems for these groups, and then show them what ethnic group they belonged in. This all began in roughly 1923, after the Civil War had ended and the Bolsheviks had consolidated their power. This was referred to as korenizatsiia (коренизация), which is derived from the word meaning to place roots (so establishing oneself).

As part of helping to develop these groups, they were given favourable positions within government and schools and so on. Terry Martin, one of the leading historians on the subject of korenizatsiia, called his book The Affirmative Action Empire, and the parallels are arguably striking (disclosure: I'm a firm supporter of his theories, though not everyone is). Underrepresented groups (read: non-Russians) were given priority in their native regions, so in places like Abkhazia the ethnic Abkhaz were favoured for political positions and given priority on a variety of things.

In theory this was meant to help develop the national conscious of the peoples, as well as help establish themselves so they did not have to rely on Russians for everything. I say Russian because that was largely the ethnic group seen as an issue, though in the aforementioned Abkhazia and South Ossetia, Georgians were the dominant nationality (and thus Georgians were regarded as the "superior" group and downplayed; some argue this was done to cause dissent in the region, but in the course of my graduate degree I argued that was not the case).

To help bolster this as well, national territorial units were established for ethnic groups that constituted a majority population in their region. Thus the Soviet Union was divided into several smaller union republics (like Kazakhstan, Turkmenistan), and autonomous republics (like Tatarstan, Yakutia, Buryatia, in Russia). Abkhazia was one exception I'll note, as I'm familiar: though they only had about 30% of the population there Abkhazia was given autonomous status, which I'd argue was because of their support in the revolution. Within these states the titular nation (so the Tatars, Yakuts, Buriats, Abkhaz) were given even more priority in affairs, and further allowed to develop.

Korenizatsiia lasted until the mid-1930s. There is no clear reason why it ended, but several theories have been presented: by this time Stalin had consolidated his power and wanted no issues; growing unrest in Europe threatened the Soviet regime by having potential rebellious groups; but arguably the policies were not working as hoped for. National identities were growing, but it was not leading to a transition to communism, only to national identities. This was anathema to Soviet policy, so Stalin started to promote the idea of a Soviet nationality, one largely based on Russians: the Russian language was made compulsory in schools across the USSR, Russians were no longer overlooked for positions, and to be Soviet essentially became a synonym to being Russian. Of course the outbreak of the Second World War threw all that out of sort, and in the drive to unify the country Russia became the "first among equals": Stalin famously gave a toast after the end of the war "to the health of our Soviet people, and in the first place, the Russian people." Then it gets into the post-war era, which was a whole different thing that I'm not as familiar on.

For reading:

• Terry Martin, The Affirmative Action Empire: Nations and Nationalism in the Soviet Union, 1923-1939 (2001). This is largely the foundational text on the subject, and is essential for anyone looking to understand more. Like I said I'm a booster of Martin's, and while he focused on Ukraine, his theory is pretty strong across the USSR.

• Francine Hirsch, Empire of Nations: Ethnographic Knowledge and the Making of the Soviet Union (2005). This looks more at how the Bolsheviks created national identities, by examining the ethnographical and athropological work done. Hirsch doesn't agree with Martin on everything, and it serves as a good counterpart to his work.

• Adrienne Lynn Edgar, Tribal Nation: The Making of Soviet Turkmenistan (2006): A case study utilising the information gleaned from the ethnographers and korenizatsiia, Tribal Nation looks at how Turkmenistan was created, largely out of nothing, as there was no real concept of an ethnic Turkmen identity until the Bolsheviks came around.

I also want to touch on your specific questions within: did Stalin give Georgia special treatment, and did Khrushchev give Ukraine anything.

Stalin was indeed Georgian, but he was not a Georgian nationalist, and grew to loath Georgians like that. He was a committed communist, which meant he was an internationalist, and later on grew to adopt a Russian identity. Lenin famously called him a "Great Russian chauvinist" and noted that Stalin's non-Russian heritage made him that much worse, as he was overcompensating. Throughout Stalin's era, it was actually his lieutenant Lavrentiy Beria who favoured Georgia. Beria was also Georgian, though notably he was not ethnically Georgian: he was Mingrelian, an ethnic group that is closely related to Georgians and from the country, and to most observers is just a sub-group, but to Georgians it is a major distinction. Beria actually favoured Mingrelians, and saw them appointed to high positions whenever he could, though he also helped bring up Georgians when he could. That is not to say Stalin didn't forget his roots: he would famously have fruit and wine shipped up from Georgia, and talked to Beria in Georgian often, in order to keep things from observers (few non-Georgians speak the language).

Khrushchev famously ceded the Crimea to Ukraine in 1954, transferring it out of Russia to commemorate the 300th anniversary of the Treaty of Pereyaslav, which some argued saw Ukraine brought under Russian control (it's more complicated than that). This was a largely symbolic gesture: it was shifted from one union state to another, but as both were subservient to the overall USSR it was irrelevant in the big picture. It could be argued that Khrushchev did this as he was from Ukraine, but seeing how he was ethnically Russian and Crimea was also largely comprised of ethnic Russians (the Crimean Tatars were deported in 1944 to Central Asia), it seems immaterial. This of course would not be an issue until 1991 when the Soviet Union collapsed and suddenly Crimea was in a different country from Russia, and even more of an issue in 2014 when Russia annexed the region.