Lately, I've noticed these very strikingly relatable things when Persia is mentioned being ruled by a foreign empire. From as far back as Alexander the Great who had such an enormous amount of admiration towards Persia. He picked Persian advisors, adopting Persian customs and generally viewing Persian culture so highly despite the fact that he was perhaps living during the time of great Greek thinkers. You would also notice how, even during the the Mughal and Ottoman Empires, would Persian culture makes its way into fruition.
When I hear about things like this, I am reminded of China's cultural significance, when a foreign power conquers China, that said conqueror, would eventually go through a process called Sinicization.
With a lot of similarities between the two when it came to cultural significance and how much, "Persianization" would go about in neighboring countries or empires that conquers Persia. Was Persia generally speaking, the China of the Middle East?
Yes, this is very much an appropriate observation! Similar things can be said about China as you have noted, and also about Greece and Italy.
As I am a historian of Turkey, I know more about the Persian influence on Turkish culture and language, so I'll tell you that story. By any estimate, there are at least around 1.000 to 1.500 Persian loan words in modern Turkish. I would imagine that a non-negligible amount of Arabic loan words must be indirect loans from Persian too. This, despite the fact that 1 - ordinary Turks of any age used more Turkic words than the elites who tended to use a sophisticated dialect of Turkish with many Persian and Arabic loanwords, and 2- in the early years of the Turkish Republic as well as later on (though not today) many Persian words were replaced by Turkic or newly derived European and/or Turkic inspired words. The Ottoman sultans were called padişah and often wrote poetry in Persian (though not exclusively), to give an example of all this. Until the 20th century, I would say, a typical Ottoman intellectual would be expected to read Persian and Arabic even before they read French or what have you. Mind you, the position of Arabic is all to be expected considering these intellectuals would, after all, be expected to read the Quran and hadiths. But Persian is not the language of Islam; it was perceived to be a language of a related high culture. That said, interestingly enough, most Turkish words pertaining to the ordinary vocabulary of Islam are of Persian origin. The Arabs call the Muslim prayer salat whereas Turks call it namaz, just like the Persian. Similarly, the Arabs call 'prophet' nabi while it is mainly peygamber in Turkish and Persian. The last one is true for other Turkic languages such as Uzbek and Azeri. Examples can be enumerated but I think this suffices to show that indeed, as you have suspected, the culture of Iran exerted a strong influence on neighbouring cultures for a very long time.
Now, Ottomans never conquered Iran. Not that they did not try though... Their rival in regional domination was the Safavid Empire in the 16th century, which was founded by Shah Ismail, an ethnic Turk/Azeri (not that much of a difference back then, frankly). The Safavid were, in time, replaced by another dynasty of Turkic origin, the Qajars. If you go back instead, there were Seljuks, who were, again, of Turkic origin. What you have is an almost continuous rule of Turkic dynasties in Iran until the 20th century. And yet, the state language of Iran remained Persian. Interestingly, for instance, the Seljuks' army most certainly spoke and wrote in Turkish. But not the bureaucracy (unlike the Ottomans). Often enough, the viziers themselves were Persian (like the famous Nizam al-Mulk from the 11th century, who was assassinated by the Hashashins), and not surprisingly, the state mechanism did show important influences of the Persian tradition. I think this is very Iran becomes most like China; being conquered by outsiders and then, almost transforming the outsiders into traditional Chinese/Persian rulers.
A word of caution in order. Though I imagine the Persian influence on Turkish is more widespread, this Encyclopedia Iranica entry provides some examples of the influence working the other way around. Similarly, when a good number of Romans were lovers of the Greek culture, there were traditionalists like Cato the Elder who voiced dislike for 'soft' behaviour of the Greeks. Later on, after the Roman conquest of Greece, then some Latin loanwords appeared in Greek. Since society is never of one mind like an individual, these kinds of things are to be expected. Seeing any influence should not lead to an expectation of full-scale borrowing or copying, which, I believe, almost never happens anywhere in the world.
This was rather sketchy but it is so largely because your question has already covered much ground! Thank you for this lovely question and hope I was able to provide some further examples and confirmation.
I am reminded of China's cultural significance, when a foreign power conquers China, that said conqueror, would eventually go through a process called Sinicization.
I think it's important to bring up that 'Sinicisation' is basically a mirage. It was a mirage originally constructed by the conquest dynasties at the time, who very rarely had to actually compromise on their own existing identities, but merely needed to present themselves as acculturated, at least in areas where it counted. It has only been with the rise of imperialistic nationalist sentiments in the early part of the 20th century that Han Chinese people have claimed that actual successful 'Sinicisation' of foreign conquerors took place historically, partly as a means of justifying the retention of Qing-era imperial holdings. I've written on this for the Qing here and here, and /u/cthulhushrugged for the Tang here.