The religious question was also central and not simply a cover.

Orlando Figes, a famous historian wrote a fabulous book on the subject(Crimea: The Last Crusade) . Yet for all its modernity, the fighting and the goals were also a “holy war” for each belligerent power. Leaders used religious rhetoric and ordinary soldiers and sailors said their prayers as they tried to make sense of what they were doing. That, presumably, His book reveals the strange mixture of meanings the war had for its combatants. He puts the conflict into its broader context: the determination of Britain (and with some reservations, France) to stem Russian expansion and to bolster Islam in its fight with eastern Christianity.

No, that last point is not a mistake. The great historical paradox of the Crimean war—and of the longer-term Russo-Turkish conflict of which it was one episode—is that Anglican England and Roman Catholic France were aligned with Islam's sultan-caliph against the tsars who saw themselves as the world's last truly Christian emperors. Above all, the western Christian powers were determined to avoid any reversal of the Muslim conquest of Istanbul: “The Russians shall not have Constantinople” chorused an English music-hall song.

How did the various players in this strange religious game explain themselves to their own pious subjects? For the theocracies of Russia and Turkey, and their God-fearing soldiers, things were fairly straightforward: they were fighting, respectively, for Christianity and Islam.

It was harder, you might think, for the Church of England and the Catholic establishment in France to explain their support of the caliphate. In fact, they found it easy enough to construct the necessary arguments. First, British and French clerics demonised Russian Orthodoxy as a semi-pagan creed. Second, they maintained that in some peculiar way the Ottoman empire was more friendly to its Christian subjects than the tsar was. (The Ottomans tolerated Protestant missionaries, so long as the evangelisers limited their search for souls to Orthodox Christians.)

In the spring of 1854, as the Crimean fighting began in earnest, an Anglican cleric declared that Russian Orthodoxy was as “impure, demoralising, and intolerant as popery itself”. What could be more natural, then, than to team up with Islam and popery to cleanse that terrible impurity? A French newspaper, meanwhile, gave warning that the Russians represented a special menace to all Catholics because “they hope to convert us to their heresy”. The wife of Napoleon III(Eugénie de Montijo empress of France) a devout Catholic gave herself the mission to demonstrate to the greatest gentlemen of France the holiness of this war.

For further reading:

Orlando ,finges:Crimea: The Last Crusade. 575 pages

Jelavich, Barbara : Russia's Balkan Entanglements, 1806–1914.204 pages

The conflict that actually instigated it was the Ottomans ignoring a previous treaty with the Russians that promised Russia the role of protectorate of the Christians in the region. The French role as an instigator is often under-appreciated. Diplomatically France had been isolated in throughout the 19th century as a result of the Napoleonic Era. They had no natural allies and had fought the entire continent and won. They were also a source of liberalism since the revolution and this made them extremely dangerous to Austria and Russia. This idea of France needing to be isolated intensified when Napoleon III seized power as an emperor in 1852, suggesting another era of Bonaparte aggression. It is in this context, along with becoming increasingly unpopular domestically that Napoleon looked abroad to help restore France's prestige. He decided to feed on russophobia found throughout Europe as a reason to incite a conflict.

Russia demanded a reversal of the Ottoman ruling and threatened war. This in turn terrified Britain and the rest of Europe as they feared Russia gaining influence in the decaying Ottoman Empire. British officials especially were scared of Russia moving into Iran (which they did) and potentially being able to reach British India. This importantly allowed France and Britain's interests to align, a rare occurrence at this point.

As a sort of aftermath the Crimean War did allow France to break from isolationism. The conference ending the war was held in Paris, symbolizing France's return as the home of diplomacy in Europe (ex. lingua franca etc.) and shifted much of the wariness towards themselves to Russia. The Ottoman Empire proved to be as weak as feared increasing russophobia.

Sources: F.R. Bridge The Great Powers and the European States System 1814-1914

G. Wawro Warfare and Society in Europe, 1792-1914

edit: Reread your question and realized i didn't focus as much on the eastern concerns that you inquired about. Round 2

The Black Sea fleet was essential to Russia as it was their access to a warm water port and the Med through the Bosphorus. The Russian grand scheme was to be the first one to take advantage of the Ottoman Empire if it was to fall and more specifically seize Constantinople to control the waterways. The pan-slavism movement hadn't begun in earnest but there were some important people suggesting it, but more so Russian religious officials were pressuring the Tsar into intervention on behalf of their orthodox brotherhood. Hope that helps.

As for why the Russians need a Black Sea fleet, the answer comes down to geography. Since at least the reign of Peter the Great, Russia looked to secure warm-water ports, particularly with the Mediterranean, and later the Pacific. A port on the Black Sea would have allowed Russia to project themselves much further than previously possible, opening up significant economic opportunities with Europe.

Besides everything already being said, there was one particular reason for France to join this war - personal relations between Nikolas I and Louis-Napoleon. Louis-Napoleon declared himself Emperor of France in December 1852, and Nicolas refused to adress him "my brother" (i.e. doesn't recognized him as an equal).

So for Napoleon III it was rather personal matter.