1. They did! It was called Byzantium. It was one of the older Greek cities, existing for much of what we remember as classical Greece (think Athens and Sparta). Byzantium was a fairly "successful," city, being on major trade routes, it just wasn't Rome or Syracuse level. Once it became the center of the government's attention, they could invest so much money that it became the metropolis it became.

2. Constantinople was the capital of the Ottoman Empire, full of Turkish people (unlike most of mainland Greece, Constantinople has a huge Turkish population at this point), and as such the Ottoman Empire was absolutely never going to give it up. Alongside this, the powers at be (Namely France and Britain) were content to let the Ottomans hold on to it. The Ottomans were weak and would yield to the demands of the French and British, while the Russians or a Russian satellite state would not.

3)Again, because of the importance of the city. The British were not fans of anyone they couldn't control running the city, which is why it was established as an international control zone, and the British wouldn't leave the city until 1923 when the Treaty of Lausanne was signed, which restored full Turkish civilian control to what is now Turkey. Alongside this, the Grand National Assembly (the rump state in control of Central Anatolia after WWI) had overrun the territories that HAD been given to Greece (Smyrna and the Ionian coast), beating the Greeks in the process, so they were in no position to try and take the city.

1. They did: Byzantion, founded in the 7th century BC, is said to gave been the greatest city of Thrace (the region north-east of Greece) in later centuries. The earlier city was largely rebuilt in the early 3rd century and seems to have been left in place when Constantine built his new city to its south and west 120 years later.

2. The Greeks had only been saved by the intervention of the Powers and were in no position to extend their rule so far. Britain in particular was opposed to a carve-up of the Ottoman territories, which indeed was seen as opening the door to Russia. Greece wouldn't even get Thessaly for another half-century.

3. Istanbul remained a strongly multi-ethnic city but far short of being Greek (official figures in the Empire's last decades claiming about half Muslim and a fifth Greek). The plan was to internationalise it while the Allies carved out their respective spheres of influence in Anatolia: even international control was too much for Turkish nationalists, and Greece's disastrous defeat in 1922 put paid to the scheme.

The problem is that as the Allies discovered in 1920 it was one thing to assign territories on paper, another in practice. At the end none of the powers was willing to commit to another war to uphold what had been a rather discreditable peace.