Is it possible for Mongols to migrate and settle in the Middle East, similar to how Turks moved to Anatolia? How would this affect the development of the region throughout history?
The Turkmenli region in Iraq would be a good place for Mongol settlement.
You have the Hazara in Afghanistan and the Kalmyks in Russia. The former were descendants of Genghis Khan's soldiers there and they used to mostly speak a Mongolian language before converting to Persian for use. It's certainly possible that you would have a large enough of Mongols to settle down and somehow not disappear through assimilation with the majority cultures. The Turkmenli region in Iraq would be a good place for Mongol settlement.
By the way from what I know there are insignificant tiny Mongol ethnic minorities as enclaves far from Mongolia proper in China (near Tibet if I am not mistaken) which are the descendants of the Mongol garrisons. And they also partially kept their Mongol language and identity almost till our days. And here again the Mongols were not surrounded by the Turkic tribes.