Greetings! This response will cover in more-depth the geopolitical perceptions of the Russian Empire in the 19th century (specifically from about 1830 to 1899) and within that category focus more on how two key powers viewed Russia: Austria-Hungary and the British Empire. Note that parts of this response have been adapted from an earlier one on relations between Austria and Russia in the 1800s, and I hope that other contributors can chime in on a wider variety of opinions about the Russian empire. Let's begin.
The Lion and the Bear
British politicians for much of the 19th century viewed Imperial Russia as a constant threat from the east in the security of their Empire. Whilst Russia could do no harm militarily to the Home Islands (given her lack of naval technology and economic backwardness), they were most concerned about what Russian expansion eastward meant for the British Raj on the Indian subcontinent. In the 1820s Russia had begun expanding downwards into the Back Sea and into dangerous orbit with the Ottoman Empire (then referred to colloquially as the "Sublime Porte" after the main government offices in Constantinople). The British feared that the Ottoman Empire's weakening grip over its Eastern European provinces in the Balkans and the open rebellion of its Arab regions (sparked after viceroy Mehmet Ali's revolt in Egypt) would give Russia the means to expand further into the Levant and control Anatolia. This was an unacceptable eventuality, as that meant Russia would exert massive (if not the most) influence over Persia, the eastern Mediterranean, and directly threaten the Suez canal. Foreign Secretary Lord Palmerston described these fears aptly in 1835:
"I take [Tsar] Nicholas [I] to be ambitious, bent upon great schemes, determined to make extensive additions to his dominions and, animated by the same hatred to England which was felt by Napoleon."^(1)
Palmerston, echoing a fear shared by many of Britain's imperial agents, feared that Russian encroachment into Central Asia would inevitably lead to conflict over the British Raj, threatening that "jewel in the crown of the Empire". To that end, the British sought to ensure that Russia's influence in Central Asia was limited (or even rolled back) from places such as Afghanistan and Persia. As Palmerston himself rhetorically phrased it:
"Sooner or later, the Cossack and the Sepoy [Indian soldier serving under Britain], the man from the Baltic and he from the British islands will meet in the centre of Asia. It should be our business to make sure that the meeting is as far off from our Indian possessions as may be convenient."^(2)
With the Russian defeat at the hands of an Anglo-French-Turkish coalition in the Crimean War, this seemed a plausible route. The resulting series of geopolitical conflicts in Central Asia however, known as the Great Game, would cost Britain and Russia dearly. By 1870, London had to accept that St. Petersburg was not budging from its place in Asia, but also that it posed no dire or immediate threat to the security of the British Raj.
The Roaring Bear and the Double-Sided Eagles
Adapted from the earlier response linked in the introduction
In 1853 we find the origins of the Crimean War as the first major upheaval in Austro-Hungarian-Russian relations and the decline of positive perceptions of Russia in Vienna. The occupation by Russia of the Danubian Principalities shortly after the debacle over the rights of Christians in the Holy Land sparked a series of negotiations between the powers of Europe. The Austro-Hungarian Empire was quick to react to the occupation in particular, joining with Britain and France in drafting the Vienna Note, which (without getting too far into its technicalities and interpretations), accepted the legitimate claims of Russia to the Ottoman-Christian territories, without overriding the authority of the Sultan. Tsar Nicholas I accepted this offer, but the Turks demanded certain changes which would have made the final Note unacceptable to the Russians.
This diplomatic back-and-forth went on for a while, until the Western Powers were persuaded to (with the supposed backing of Austria-Hungary) issue an ultimatum that the Russian immediately leave the Danubian Principalities by April 30th, 1954. When the tsar made no reply and the Habsburg support turned out to be merely diplomatic rather than militarily, Britain and France declared war on March 27th. It is this "big picture" narrative of events that we must keep in mind, as we pick out the specifics of the Austro-Hungarian Empire's role in the negotiations.
The Habsburg monarchy was a critical consideration for the policy-makers in St. Petersburg, including the Tsar himself. To that end, Count Orlov (a prominent diplomat) was dispatched to Vienna in January 1854 in order to secure Austrian neutrality for the war which was predicted to come (the Russian minister in Berlin was ordered to do the same for the Prussians). So concerned with the intervention of Prussia and Austria-Hungary was Russia, that Orlov was authorised to guarantee the integrity of Austria-Hungary from French hostilities. Knowing full well that such negotiations would add another complex dimension to the overarching ones with Britain, France, and the Ottomans, a young Emperor Franz Joseph declined to talk about such matters. Both he and the Austrian Empire's foreign minister Count Karl Ferdinand von Buol insisted instead that Russian troops not cross the Danube, for that would incite further reason for Austria to side with the Western Powers.
Mind you, the Western Powers were also finding it rather difficult to deal with Vienna in their own diplomatic efforts. It would not be an overstatement to suggest that the opinion of the Austro-Hungarian Empire quite literally was the lynchpin of the entire negotiation process. This was particularly evident in the surprise of the Western Powers when Austrian support for the ultimatum turned out to be merely diplomatic in nature. Historian Bernadotte E. Schmitt writing in 1919 about this concern notes:
"They presented their ultimatum [for the evacuation of the Principalities] on the understanding that Austria would support it, whereas she actually gave only diplomatic approval, not a promise of military assistance, which alone would have compelled Russia to yield."^(3)
The difficulty for the Austrians was considerable. The memory of Russia's intervention (and specifically that of Tsar Nicholas I) during the Crisis of 1848 remained in the minds of both Emperor Franz Joseph (who came to power largely in part thanks to Russian assistance) and top-level advisors across the Empire. However, all of them also feared that if Russia maintained dominance in the Danubian Principalities, she would set up vassal states whose Slavic majorities would lead to further complications with the Slavic populations living under the Dual Monarchy. On the other hand, these leaders were also reluctant to cooperate with Britain or France in their opposition to Russia. The former was viewed as the home of liberalism and the home-in-exile of Hungarian lawyer Lajos Kossuth, who had played a considerable part in the 1848-9 revolution. The latter, ruled by Napoleon III, was viewed as the champion of Italian independence, which upset Austro-Hungarian rule in the region.
In the end, Austria became increasingly hostile towards her old ally in the east. After Count Orlov admitted that Russia would indeed set up vassal states in the Balkans, the Emperor insisted that they now only conduct military campaigns on the Asiatic side of the Ottoman Empire (an impossibility given that the main crux of hostilities lay in Eastern Europe). Even as the Russian troops crossed the Danube on March 22nd, the Austrians were reluctant to place their army on high-alert.
Just before Austrian troops were put on high-alert and the Russian crossed the Danube, the Habsburg Monarchy managed to secure an alliance with the Prussians. This alliance never had to wield its guns in the Crimean War, but it did set a precedent of thinking in Vienna which would characterise the next several decades of Russian antagonism by the Austro-Hungarians.