Allies?

The Byzantines were assisted several times by the Venetians and Genoans, who knew that if Constantinople were to fall, that their trade would be hurt severely, as all trade flowed from the East. And in fact after the conquest of The City in 1453, the Ottomans increased the tax for "Dar al-Harb" (people who live outside of the realms of Islam) merchants, focusing on internal trade and revenue, rather than international commerce. This is not to say that they cut off trade with the Venetians or Genoans, but rather that they were not going to be pushed around to grant trade concessions like the Byzantines had previously.

EDIT: Look up the Venetian- Safavid Persian and Venetian - Mameluk Egyptian alliances to counter the Ottomans. These were attempts to limit and counteract the growing influence and power of the Ottoman Empire in the Eastern Mediterranean, which effectively came to an end at the Battle of Chaldiran 1514 against Shah Ismail (Safavids) and the 1517 conquest of Syria (al-Sham) and Egypt by the Ottomans.

In fact, where as the Venetians did not end up sending any assistance to aid Emperor Constantine XI, but rather it was the Genoans, who fought alongside the Emperor and the last defenders of the city. Again this is not because of some moral obligation (or spiritual fraternity) on the behalf of the Genoans, but because they had significant trade interests in the region (see Tower of Galata), and the fall of the city would mean their quick decline in the Black Sea region.

Important to note that this was all before Vasco de Gama's discovery of India (around the Cape of Good Hope) and before Cristobal Colon's 1492 journey to the Americas.

As for other actors, such as the Russians, you'll have to ask a Russian historian on what was going on with Muscovy in 1453, but they were truly outraged, but effectively could not do anything about it, and it eventually served their purposes as they took on the leading role in the Orthodox community. As for the French and English, who the Emperor Constantine XI's father had visited, they were still busy fighting the end stages of the 100 years war. Although they expressed moral support, after 117 years of fighting each other on and off, a devastating plague, and with most of the war being fought on French land, neither France nor England were in any position to assist the beleaguered Byzantium.

The Pope was on the side of the Byzantines (the Emperor who had converted to Catholicism to win his support), but the Pope only has the power of the pen, and relies on the power of the men with armies, to enforce the pen, and hence with Crusading glory having already been vanquished at the failed Varna Crusade (against the Ottomans), the Pope's moral authority had been left asunder, and no one interested in Papal crusades.

Roman Rebels Post-1492?

As for rebels trying to restore the Byzantine Empire, as far as I know (taking a class on Ottoman History right now), the answer is no, up until the era of nationalism, and the independent Greek "Megali Idea", or Greater Greece, which would have restored Greece or Byzantium, to its 'original borders'. This however was soundly defeated by Kemal Mustafa (Ataturk), during the Turkish-Greek Wars in the post-WW1 years.

You have to understand, that the Ottoman Turks did a very good job of eliminating pretenders to not only their throne, but to the title of Byzantine "Roman Emperor", and immediately after the conquest of Constantinople, made sure all claimants to the throne were dead. The Ottomans, at least early on, also did a good job of co-opting Greeks into their administration, as well as other minority groups, as at the end of the day the Ottoman Empire was a "Balkan Empire".

Source:

Lars Brownworth, "Lost to the West"

"The Fall of Constantinople to the Ottomans", Michael Angold

If anyone has any further questions just let me know (I enjoy procrastinating writing a history paper, by writing another one :P )

Well, at the final siege of Constantinople the defenders were a pretty multinational force! They consisted mostly of native Greeks with a respectable amount of Genoese and Venetians (who had significant trading interests in the city). Furthermore Constantinople hosted a pretender to the Turkish throne (the Greeks liked to have one on hand any time they needed to shake things up with the Ottomans) who fought to the death. Additionally, there was a Catholic bishop(?) and a few hundred soldiers who had been sent to ensure the Schism was properly ended (the Greek leaders agreed to unite the churches as a desperate final measure). And last but certainly not least, there was a lone Scotsman! His name was John Grant and he was instrumental in keeping the Turks (who were quite the multinational force themselves) from tunneling under the walls.

(EDIT: How could I forget, there was a small Catalan unit fighting there too, thanks /u/Toc_a_Somaten)

Giovanni Giustiniani, a Genoan, was arguably just as important a leader as Emperor Constantine himself. He led 700 soldiers to Constantinople to aid in a defense that was nearly hopeless. But Giustiniani never faltered, he was a brilliant, inspiring siege commander who kept all the different factions of Byzantium together.

Calling it a single "war" against the Ottomans would be problematic for answering your question. The Ottoman dynasty arose in Anatolia ~1300 and it wasn't until 1453 that the ERE finally fell.

As for your second question, the Greco-Turkish War of 1919 comes to mind. The Greeks were fueled in large part by nationalism and wanted to take back Constantinople. They failed, of course.

Follow up: At what point did Turkish culture (As opposed to Greek) and Turks as an ethnic group become dominant in modern day Turkey? Was that always the case in Byzantine history? Was the fall of Byzantine and rise of Ottoman primarily because of a cultural shift in Turkey?

There was a catalan garrison defending constantinople during the siege, they defended the Bukoleon Palace area in the south west of the city, near the sea, their commander was Pere Julià, and he, along with the catalan consul (a kind of plenipotenciary commercial ambassador), Joan de la Vila, were executed when the city fell and they refused to flee. The fall of constantinople had a big impact in Catalonia (as in most of europe), with the addendum that the catalan royal house had family links with the older ruling house, the Komnenos (Eudoxia Kommenos was the grandmother of James the Conqueror)

Regarding your second question, the simple answer is that there was no significant attempt. Put simply the Ottoman empire was too big and by the fall of Constantinople, Greek controlled lands were small and would be absorbed over the next decade.

General practice for the ERE (and modern historians) was that to be Emperor you needed to control Constantinople (emperors were crowned in Hagia Sophia). Of course, once it became the Ottoman capital, and turkfied that became effectively impossible.

There were still other claims to the title. There was an 'Emperor of the Greeks' in Trebizond dating from before the crusade, who was conquered in 1463. Constantine's brothers were still vassals of the Ottomans in the Morea, although (sensibly) neither claimed to inherit the emperorship. Demetrios surrendered his lands and Thomas fled to Italy in 1460 after Murad had to intervene in a war between them, and simultaneous revolt against them. Thomas's son Andreas would claim the title in exile (with effectively no justification) and later sold it to the King of France. His death would also end the male like of the Palaiologi. The Grand Duke of Muscovy married his sister Sophia and this was used by Russia as a claim on Constantinople in time.

The Greeks under the Ottomans were not generally rebellious until the 1800's. Early on, this was largely because the Ottomans retained the Orthodox Patriarchate, and stopped the unpopular union with the Catholic Church. They also generally respected the Greek nobility and clergy. At least some elements of Greek nationalism post 1800 favored a restoration of the Byzantine empire.

Sources: Jonathan Harris: "The end of Byzantium" and John Jules Norwich "Byzantium" (although regrettably, at this point he uses flawed sources). That said there are a large number of books discussing the siege and its consequences.

They were somewhat allied with the Catholic West, but they're the ones who screwed them over.