The non-Russian territories can cover a wide region of of space and the imperial state's management of them varied greatly over time. For this answer, let's focus on the three the question (Ukraine, Belarus, Georgia) mention and limit the timeframe to the nineteenth century.

Ukraine.

The Russian imperial state tried and failed to assimilate its Ukrainian population into an “All-Russian nation” over the course of the nineteenth century. In short, the imperial state saw Ukraine as a "lost" Great Russian people and the empire was the means to bring them back into the fold. According to Alexei Miller, the failure to unify Ukrainians and Great Russians into a cohesive whole was in no small measure due to the uneasy relationship Great Russian nationalism enjoyed with the official nationalism embraced by the autocratic state. The latter implied that the Romanovs would accommodate a certain level of ethnic particularlism provided it strengthened the empire. While Great Russian nationalists largely supported any project that strengthened the state, they were highly skeptical of giving Ukrainian activists too much freedom. The Russian journalist Mikhail Katkov typified the incompatibility of Russian nationalism with imperial cosmopolitanism. Katkov's editorials forcefully argued that liberal accommodation of Ukrainian language acted as a Trojan horse for the Polonization and eventual loss of the territory. Anti-Ukrainian language laws like the Valuev circular and the Ems Edict were counterproductive. The attempts to limit the written Ukrainian language through both the Valuev Circular and the Ems Edict, while not examples of a malicious chauvinism, were ultimately counterproductive as it inadvertently politicized cultural activity of Ukrainians.

Belarus

Unlike Ukraine, the "All Russian" project had to greater success here. Like Ukraine, the imperial state projected onto Belarus a lost Russian identity that the state would reclaim. This was a mixture of willful ignorance and wishful thinking. A salient feature of the late Romanov state was the marked discrepancy between its expansive aims and the structural limits a multiethnic empire imposed upon those aims. The poverty of the Russian administrative apparatus, both in resources and personnel, meant that they had to rely upon scientific institutions like the Imperial Russian Geographical Society (IRGS) with mixed results. For example, the IRGS’s ethnic maps of Belarus often ignored cultural and religious incongruities to make this part of the Northwest consistent with that state’s expectation that this region was “Lithuanian-Russian” (itself an invented term) and thus a safe imperial zone. Within the Northwest Provinces (Belarus and the Baltics), imperial policy aimed to assimilate those it deemed Belorussian, acculturate its Jews, and eliminate any perceived vestiges of Polishness. Confessional identity quickly emerged as convenient shorthand for imperial officials to project a fictive homogeneity. Belorussians accepted this imperial project to a degree. Unlike Ukraine, there were other ethnic rivals (Lithuanians, Latvians, Baltic Germans) within the Northwest and Belorussian nationalist activists (the few that there were) saw a tactical advantage with being associated with Great Russians.

Georgia

The case of Georgia is reflective of the highly heterogeneous nature of Russian imperial expansion. There often was no singular plan or set policy for the peripheries. When the Russian state annexed and conquered Transcaucasia in a series of wars during the early 1800s, the Russian state promised the Georgian nobility they would receive the same status as Russian nobles. The Georgian church though did not have the same level of accommodation and was forcibly integrated into Russian Orthodoxy. The tsarist empire tried to Russify Transcaucasia through a number of settlement schemes. For example. Nicholas I exiled the Doukhobors (religious dissenters) to Transcaucasia between 1830-41. The rationale was that not only would this remove the Doukhobors' spiritual contamination of Orthodox Russian peasants, but also service on this frontier area would transform the Doukhobors into Russians. This project failed as the Doukhobors formed a unique community within Transcaucasia. Additionally, the continued presence of hereditary Georgian nobility meant that the nascent Georgian nationalist activism had an educated leadership caste. The absorption of the Georgian church into Orthodoxy lessened religion as an criterion of Georgian national identity. By the twentieth century, the Georgian nationalist movement was split between an elite-centered movement that could cooperate with the state, and a larger, more politically radical group among the emerging Georgian cities.

Sources

Breyfogle, Nicholas B. Heretics and Colonizers: Forging Russia's Empire in the South Caucasus. Ithaca: Cornell University Press, 2005.

Hillis, Faith. Children of Rusʹ: Right-Bank Ukraine and the Invention of a Russian Nation. 2013.

Kappeler, Andreas. The Russian Empire: a multiethnic history. 2001.

Miller, A. I. The Ukrainian Question The Russian Empire and Nationalism in the Nineteenth Century. Budapest: Central European University Press, 2003.

Staliunas, Darius. Making Russians Meaning and Practice of Russification in Lithuania and Belarus After 1863. Amsterdam: Rodopi, 2007.

When Ukraine, Georgia, and Belarus, for example, belonged to the Soviet Union, they were managed as semi-independent nations.

This is not really true. There were less powers of independence and autonomy in the Soviet republics than there are for US states. The USSR was a pretty monolithic entity for most of its history.

Russian Empire was managed differently depending on the time period and political situation. The Empire was divided into Gubernyas (Governorate), that were divided into Oblasts (Regions). Here's a good pic from Wiki on how it was subdivided just prior to the revolution. The way the divisions worked and how they were managed changed very frequently. Peter the Great made a bunch of reforms, than Catherine I changed them back, Catherine II the Great, made her own, then there were a slew in the 19th century, with many coming as a result of the Abolishion of Serfdom in 1861, while other changes were due to conquest and expansion of the Empire. For example Derbentskaya Gubernya (Daghestan) was remade into Daghestanskaya Oblast and governed by a Military Governor, and Byalstok Oblast was given to Prussia in 1795 and given back to Russia in 1807 as part of the Treaties of Tilsit (Alexander I and Napoleon).

The structure of management and autonomy or lack thereof changed drastically in 1905 as a result of the 1905 revolution, and of course the Soviets made their own changes.

As a follow-up: were dialects of Russian outside of Moscow and St. Petersburg any less different from standard Russian than Belorussian is today?

In Martin MacCauley's "Rise and Fall of the Soviet Union", he describes how monolithic the Soviet system was. Official Soviet policy was complete homogenization of the Soviet system throughout the USSR with no regard for nationhood or ethnicity. It was believed to be a non-biased system, a shining example of how communism overcomes bigotry. This dogma required in all official works and no one dared contradict it.

Social scientists and ethnicity experts had to maintain the party line or face severe repercussions. Their "discoveries" had to comport with Marxist/Leninist ideology and demonstrate how the Soviet system either invented or perfected the positive results described in their studies. And yes, those studies had better be positive. If someone dared to infer that citizens of Tajikistan were being systematically deprived, that person was not likely to enjoy life outside a gulag for very long.

In reality, of course, places like Tajikistan got nowhere near the support and attention of Ukraine or Belarus. This reality was painted over by successful efforts to enforce control over language, economy, religion, and the expectations of daily life. All citizens felt a sense of camaraderie with Russia, which was viewed as "Best Soviet Republic".

Doing things the Moscow Way was paramount and no amount of nationalism or national pride was tolerated. The republics were mere borders delineating historic regions that no longer applied. Fandorin is correct in that individual states in the USA had more autonomy and cultural quirks than republics did in the USSR.

Many more countries were in soviet union. I am from one of them. Russia occupied these countries and exiled people to syberia to die or just killed them. Everyone who was not russian and had brains was killed or persecuted. That's why now we have such dumb society, soviet heritage(stupid, alcoholic, fascist people). Most of the clever ones were killed by russians and the dumb ones multiplied. No one was happy in soviet union, it's all was a propaganda. We were not happy, believe me everyone wanted out of that poverty and red fascism, only retards liked it. And one day there was enough of normal people to fight for freedom from that red scum. Russia now falsificates history facts about ww2 and soviet unions, they lie that stalin was a hero when he was same as bad as hitler and even killed more people. Russians still praise stalin. And now russia became a fascist-terrorism supporting state.