The answer to all those questions is pretty much: they did! The Treaty of Sevres was rejected by Ataturk and lead to the Turkish War of Independence where they fought the French, Italians and British, as well as local enemies like the Greeks and Armenians.

The European powers didn't have much appetite for another long war at this point, and the Turks won, with the Treaty of Lausanne in 1923 settling (roughly) Turkey's modern borders. Lausanne had a few vagaries to it though, which have actually been cropping up in the news recently. An ambiguous Turkish claim to Mosul from Lausanne was brought up by Erdogan to justify his military and political maneuvers in the region, though this is another story and one outside of the 20 year historical scope of this sub.