As far as I'm aware the letter is not historical. Unfortunately I can't find a reference which comments on it, but Victor Ostapchuk makes no mention of it at all in his analysis of Ottoman-Cossack diplomatic relations between 1648 and 1681. ("Cossack Ukraine in and out of Ottoman Orbit, 1648-1681," in Gabor Karman and Lovro Kuncevic eds, The European Tributary States of the Ottoman Empire in the Sixteenth and Seventeenth Centuries (2013). The language and terminology in the letter (as it appears on Wikipedia, anyway) does not seem to me to be typical of the Ottoman style. Phrases like "son of Muhammad; brother of the sun and moon" make no sense at all as something written by the Ottomans to describe the sultan. I've never seen the Ottomans describe Iraq as anything other than Iraq, but it's plausible that Europeans might translate "Baghdad" as "Babylon."

I've never seen any document which would verify infamous letters you mentioned (despite me being an art historian who loves Repin). I tend to think that the letter itself is a hoax (as chamboz wrote).

Regarding the relationship between Babylon and Iraq: I've never seen those words being used interchangeably. In fact, Babylon is not one of the most recurring themes in Islamic theology and folklore (Quran mentions Babylon just once [Quran 2;102] in a different concept). Here's a quick research:

Beyan-ı Menazilname-i Sefer-i Irakeyn (the chronicle of Iraqi campaign led by Suleiman the Magnificent in 1530's) written by Matrakçı Nasuh is one of the most prominent Ottoman documents regarding Iraq. He doesnn't just not use the words Iraq and Babylon interchangeably, he doesn't mention Babylon at all. Cities and villages located around ancient Babylon, mentioned by Matrakçı Nasuh are Kerbela (Karbala; كربلاء), Beriyye-i Kerbela (Countryside of Karbala), Necef (Najaf; نجف), Şehr-i Kufe (Kufa; كوفه), Beriyye-i Kufe (Countryside of Kufa), Zü'l-kifl Nebi (زالكفل نبي), Kala-i Enhare (انهاره قلعه), Hille (Hillah; حله), Han-i Beriz (خان بريض), Kala-i Berrani (قلعه براني), respectively.

Babylon and Iraq weren't used interchangeably by Evliya Çelebi in his great Seyahatname (17th century). He mentioned Babylon couple of times (in 4th, 8th and 10th books) while referring to the story from Quran (Harut & Marut) or using it as a literary tool to depict a gargantuan cliff. (Note: He didn't use the word Babylon in his Iraq-oriented 9th book of Seyahatname)