This is a difficult question for a variety of reasons.

A big one is that what was the Russian Empire went through a lot of intense events in a very short period, and we don't really have a clean tsarist "before" followed by a Soviet "after".

The Russian Empire was involved in the First World War from July 1914 until March 1918, and the end of the war would see large chunks of that empire split off, in some cases permanently (Poland and Finland were gone, the Baltics and Moldavia were gone until World War II, and Ukraine and the southern Caucasus were independent until 1920-1921, more on that in a moment). All in all that peace saw Russia lose about 40% of its population and some of its most developed regions (it lost 90% of its coal mines, for example).

Anyway, the end of the war came after not one but two revolutions: the first in February 1917 (actually March in the Gregorian calendar) and the second in October 1917 (November for ditto reasons). The first overthrew the tsar and established a republic under a very uneasy dual system of the Provisional Government and Soviets (which were not yet controlled by the Bolshevik Party), with a final constitutional settlement to be worked out by a Constituent Assembly elected in December. The second revolution was undertaken by the Bolsheviks, who had gained control of the Soviets, and overthrew the Provisional Government. They still held the Assembly elections, which they got a minority of votes in, and closed the Assembly down after one day.

Subsequent to this (to make a long) story very short, non-Bolshevik factions ended up fighting against the Bolsheviks in the Russian Civil War, which also included (I wrote a bit about this) interventions by a number of foreign countries, as well as warfare between the Bolsheviks ("Reds") against anti-Bolsheviks ("Whites"), as well as peasant rebels ("Greens") and some anarchist groups ("Blacks"), most notably Nestor Makhno's outfit in eastern Ukraine. The Reds and Whites in particular also variously fought against a number of newly independent nations, most notably in the Caucasus, Ukraine, Baltics and Poland (with the latter areas successfully maintaining their independence). The major fighting ended in 1921, but the Bolsheviks didn't even regain nominal control of areas like the Russian Far East until 1922, and insurgencies continued onwards (especially in Central Asia) well into the late 1920s and beyond.

On top of this (and largely because of this warfare), Bolshevik Russia also experienced massive de-urbanization and de-industrialization, and as well as a famine in 1921-1922, which in turn contributed to major pandemics like typhus. All in all, maybe 1.5 million died in World War I (estimates range from 1 to 2 million), maybe another million killed in the fighting and atrocities of the Civil War, and a further 5 million deaths from famine and disease in the 1920s. On top of this 1 to 2 million people are estimated to have emigrated abroad permanently (this is in addition to former Russian territories that became independent).

The Soviet Union at its formal creation in 1922 was in a deep, deep hole. Industrial output in 1921 was about 30% of what it had been in 1913, and the non-agricultural workforce had fallen from 2.6 million in 1917 to 1.2 million in 1920.

Anyway, generally speaking comparisons are made between Russia in 1913 and the USSR from 1922 afterwards, as the former is the last year of peace in the Russian Empire, and the latter is as good a starting point as any for Soviet statistics, given the war and turmoil in between. If we're looking at Soviet standards to 1929 (this is probably a better starting point than 1930, because it's the beginning of Stalin's unquestioned rule, of collectivization and the First Five Year Plan), we have to keep in mind that most of what was being undertaken was reconstruction in a very literal sense. Economic policy itself engaged in a "tactical retreat" from War Communism after 1921, and under Lenin's direction the New Economic Policy was in effect, which essentially encouraged private agricultural production and market sales - this would be ended with Collectivization and the Five Year Plans. 1925-1926 is generally considered the apogee of NEP.

In some ways the reconstruction of the now-USSR in this period meant that 1913 levels were reached and even surpassed, especially in some areas as rail transport and electricity production. Other areas of production were approaching but not yet at 1913 levels, like grain production (but with a big difference, which I will note). Other areas like coal and iron production were still well below 1913 levels.

As for grain production - we need to note that the USSR was still overwhelmingly a rural, agricultural nation in this period (and well after, until 1960 or so). The countryside in the 1920s was effectively not controlled at all by the Bolshevik Party outside of provincial towns, and peasants had more or less taken land for themselves after 1917. This actually meant that peasants were now producing produce close to 1913 in the late 1920s, but under vastly different circumstances - large landowners and even more prosperous peasant households of the pre-1917 were gone, and with them about 70% of marketed grain. Middle and poor peasants were producing more and consuming more, but less was making its way out of subsistence agriculture for urban residents to eat, let alone for the government to export. Manufactures from cities for consumer use were still recovering, and often from the perspective of peasants were too expensive to purchase. Also gone were much of the agricultural improvements that larger producers had adopted in the late 19th and early 20th centuries. Planting was mostly being done with wooden ploughs in a field-strip system reminiscent of Medieval Europe, with half the grain harvest reaped by sickle and threshed with flails. This is pretty much the setting that the discussions and debates, and finally the aggressive moves to implement collectivization and industrialization occurred in.

I suppose I've moved away from living standards to more general economic standards, but this is important context. The USSR in 1929 was still recovering from the years of war and famine that started in 1914 and ended (mostly) in 1922. The 1920s saw real improvement, albeit very uneven in terms of the economic sector and geographic location. More general statistics like life expectancy improved over the 1920s, but this was coming out of a deep hole, and was still not much above 1913 levels. Literacy was a genuine improvement - something like 25% of the population of the Russian Empire was literate in 1917, and the rate shot up to 70% in 1927 and over 90% by 1939. There were especially big improvements for women and for non-Russian nationalities, who were able to learn to read in their native languages (which were in some cases codified in written form for the first time). This was also the period of korenizatsiya or "indigenization", when Bolshevik authorities set out to promote national minority leaders for delimited territories to accommodate various nationalities. This policy would be largely reversed in the 1930s, but at the time it definitely provided a degree of cultural autonomy that many peoples in the USSR had not experienced previously.

It's all a very mixed bag. There were some major improvements in the 1920s under Soviet rule, but in many ways the country was still recovering.

There is already a fantastic overview to your primary question, so to briefly address a few additional questions you had:

I am interested in knowing what major successes the Bolsheviks had in raising living standards and in what way? Finally, would these improvements to living standards have likely happened if the revolution failed & Russia had stayed Tsarist?

Literacy rates are a major improvement of the 1920s, and with that I feel it is worth drawing out the cultural milieu of the Bolshevik era: poetry, novels, drama, and art were produced that could not have existed in the Empire, both high and low. If the last decade or so of the Empire can be characterized by the emergence of avant-garde, the Bolsheviks mark the zenith of the movement. Art and literature that were restricted to a small slice of educated society found widespread state support (e.g., the revolutionary collective Proletkult, though it was dissolved in 1920). Eisenstein's films, the war stories of Isaac Babel, Melnikov's architecture could not have been possible, or certainly as popular as they were, before the revolution.

Whether you consider art an improvement in 'living standard' is maybe murkier, but the democratization of access to (Soviet) culture is certainly worth drawing out--by the 1930s, Stalin had christened writers as 'the engineers of the human soul,' and the relatively wide latitude of the 1920s had closed in favor of Socialist Realism. The spirit that animated the Bolshevik period influenced the state sponsorship of science, both for fantastical (unsuccessful) ventures like blood rejuvenation and space exploration, and more pragmatic, modern research into agronomy, engineering, physics, and geology. Scientific research enjoyed immense state support and, at least until 1928, prioritized the specialized knowledge of Imperial intellectuals.

Counterfactual histories are interesting and challenging projects, but it is difficult to characterize whether a continuation of the ancien regime would have had a better standard of living--for the millions dead due to civil war and famine, or the for re-urbanized, newly-educated peasants?

On balance, and rather conjecturally, by most metrics I believe by 1929 life was probably better on average in 1913. In the 1930s we have the parallel threads of rapid and effective industrialization contrasted with a closing of intellectual and public life, punctuated by the purges of the later 1930s--not to mention the famines of 1932-33.