Since Mohammed was an Arab, and he wrote down the Koran, it was written in Arabic.

Farsi is a completely different language, since it's one of the Indo-European languages in some ways it's closer to English than to Arabic, which is one of the Semitic family of languages. However it is normally written in Arabic Script and there's been a lot of borrowing of words. It's spoken also in parts Afghanistan.

However the suggestion that "Iran, like many other 'nations' in that part of the world, were actually cobbled together by various victors in World War I" is rubbish. Persia as it used to be called, is one of the most ancient nations in the world, far older than the states of Europe. As with many countries the borders has changed from time to time and there are ethnic minorities.