Great question! There are plenty of things that situate the intellectuals in late Imperial Russia, but it's worth noting the sort of scope you're talking about: Tsiolkovsky's major work is right around the turn of the century, while Mendeleev was working some thirty years before. Chekhov was most active the same time as Tsiolkovsky, while Dostoyevsky was much more the contemporary of Mendeleev (and, in fact, wrote his first novel, Poor Folk, in 1845 - well before Tsiolkovsky and Chekhov were born). Tolstoy was active for much longer, but his most famous works are from the 1860s and 1870s, which I'll return to in a bit. (As an aside, Nickell's The Death of Tolstoy is a very interesting exploration of the state of the Empire just a few years before revolution). Tchaikovsky dies in 1893, while many of Rachmaninoff's most famous works were actually composed in America after he fled to Germany following the revolution.

I highlight the dates here to show that, from a handful of famous Russians, you're actually looking at over fifty years and many generations of artists and intellectuals. If we did the same for the United States, you might find a list including anyone from Stephen Crane and Orville Wright to James Audubon Washington Irving, and wonder just what was it that made America so special in the nineteenth century. There really is quite a lot happening throughout the nineteenth century in Russia, spanning many generations; the Decembrist Revolt in 1825, for example, involved quite radical notions (including freeing the serfs - Muravyov looked up to the American constitution, but rejected any form of slavery). And there are several intellectual circles in Petersburg and Moscow throughout the 1830s and, crucially, the 1840s that ally themselves with the Junghegelianer, but the revolutions of 1848, the ruinous Crimean War, and just general mismanagement under Nicholas I, faced significant political repression. These groups ultimately end up mentoring the next generation of radicals in the 1860s and 1870s, the former of which sees the freeing of the serfs.

Philosophers like Herzen, Kireevsky, and Aksakov are very invested in these questions of the place of Russia, the cultivation of Russian literature and culture, and so on; one Polish scholar argues that Russian 'self consciousness' developed in the 1840s, in contrast to the 'blind imitation' of the West. Certainly that is the impression of Petersburg that Mickiewicz records in Dziady.

The first half of the 1860s are really a highpoint in reform under Alexander II, but he pulls back some of these in 1866 and further restricts political freedom throughout the 1870s. But these underground groups simmer along (consider Chernyshevsky and the infamous summer of 1874) - Alexander is ultimately assassinated in 1881. Alexander III's reactionary, autocratic rule throughout the 1880s and early 1890s hampers intellectual circles, and he is succeeded by Nicholas II. Nicholas is an immensely complicated figure, but his balance of reform, oversight of urbanization and modernization, and disastrous Russo-Japanese War are all important backdrops to these intellectual movements.

One reason you'll find many notable figures in the 1860s is the relative freedom in the early reign of Alexander II, particularly in contrast to the 1850s. We also have the crystallization of academic disciplines and the growing internationalization of Russian institutions (not that they were particularly insular before - Peter the Great was very conscientious of that - but modern scientific fields in general aren't quite as old as you might think. Mendeleev, for example, is famous for the first periodic table) to get up to speed with German counterparts. Another point to note is that the Russian Empire has a long history, again returning to Peter I, of heavily investing in technical schools and scientific (generally applied science) education. I would point in particular to the abolition of serfdom in 1861 and industrialization over the following decades leading to a larger urban middle class, a more literate society, and a semiliterate, massive working class that is ripe for revolution. Music is a very interesting aspect, and one I am not really qualified to comment on, except to say that the 1870s to 1890s were very active. There were concerns of modernity and history, Russian exceptionalism and internationalism - these play out particularly with "The Five" and the Belyayev circle (Tchaikovsky's role here as mediator between Russian heritage and Western music is fascinating)

Russia, also, is quite large! At the 1897 census, the Empire had a population of 125 million. The United States, in 1900, had a population of 76 million; England, 41 million (and the entire British Empire 274 million); Germany, 54 million. The Russian empire had a population the size of America and Germany combined, so these figures aren't necessarily overrepresented in their international significance nor is this time period particularly exceptional in producing intellectuals.

Russia transitioned from a fairly backwards society, especially outside urban centers, to a rapidly industrializing one over a matter of decades. The Russian Empire fell from the victory of Leipzig to the boondoggle of the Crimean War, and the freeing of the serfs radically reinvented Russia's working class and landholding gentry (consider Turgenev's On the Eve). Educated Russians were well aware, too, of the changes throughout Europe, and struggled with how to relate an often imagined ancestry to the modern world.

This certainly provides a dynamic backdrop to the kinds of literature, art, and music we find beginning in the mid-19th century.

Many of the Russian intelligentsia, by the fin de siecle, joined the working classes in calls for reform. Certainly by 1911, with the assassination of Stolypin, preserving the old guard seemed impossible, and particularly experimental forms of literature, music, and art began to flourish under the decaying political situation - it's interesting that none of the avant-garde artists made their way to your list. There are dozens and dozens of rich movements and intellectual circles throughout Russia on the eve of revolution (a very, very small slice of this might be the first chapter or so of Weiner's Models of Nature - even as late as 1917, groups of ecologists and naturalists sought to establish nature reserves).

Lunacharsky, in 1928, argues that

Our literature is passing through one of the decisive moments in its development. A new life is being built in our country, and literature is learning more and more to reflect this life in its as yet undefined and unstable forms...

But I'm rather a fan of the more succinct manifesto of the Hylea circle:

Wash your hands which have touched the filthy slime of the books written by the countless Leonid Andreyevs.

All those Maxim Gorkys, Krupins, Bloks, Sologubs, Remizovs, Averchenkos, Chornys, Kuzmins, Bunins, etc. need only a dacha on the river. Such is the reward fate gives tailors.

From the heights of skyscrapers we gaze at their insignificance!