No, the Ottomans did not make use of such titles, you are correct to find them questionable! As the letter is not really discussed in Ottomanist circles, I can't delve into the the question of whether or not the famous reply was genuine (although according to Victor Ostapchuk, an expert on Ottoman-Cossack relations during this period, it is apocryphal). However, it's clear to me that at least this aspect of the story, the letter supposedly sent by the Ottomans, is not a genuine letter as it both contains many phrases which the Ottomans would never use and does not include many which would have been standard. It reads very much like someone trying to parody Ottoman diplomatic style. I've written on this before in this thread.

One thing I may add this time is that this letter (which would be classified as a tehdidname, or "threat-letter") does not follow the structure of Ottoman diplomatic style either - it should begin with (1) salutations, addressing the target and describing specifically who the letter is for, (2) describing the issue in need of resolution, including any relevant information about the recent events that prompted the letter, (3) describing how the issue can be resolved, (4) describing the magnificence of the Ottoman sultan, giving him honorifics, describing the power of his armies, and threatening consequences of failure to resolve the dispute.

This letter just opens with Ottoman titles, offers no description of why the Cossacks are being addressed, and tells them vaguely to "submit." It's written in first-person, as if the sultan himself is addressing the Cossacks, which differs from the letters that I have read, which are written from the perspective of the Ottoman state/collective. i.e. "Our Sultan commands you to..." not "I command you to..."