Ah god, this is actually a little my specialty and I'm embarrassed to tell you that I don't have an exact answer! Honestly much of your question is tied to the patterns of army raising and composition that are still unresolved topics in Ottoman historiography. The system that made the Ottomans a great Empire was the way they raised their large cavalry armies from land grants called Timars -- I believe the timariot siphahis were the powerful backbone of the empire until the death of Suleyman the Magnificent in 1566. From this period onward you see a decline of the 'classical' Ottoman state and the growth of a landed hereditary nobility with power centers not directly controlled by the Sultan and State -- and so the army transitions to use of Azabs and an expansion of the Janissaries. This century is, if I'm not mistaken, the era of the Spanish Tercio (if you're familiar with it) and into the linear pike and shot of 17th century central European style. So I would say that there is no precise dates for anything, but Archery declining (but as a traditional part of Turkish culture still vibrant) through the 2nd half of the 16th century as the timariot siphahis became less a part of the army. Considering that Napoleon's Curiasseurs were heavy cavalry that wouldnt have been super unfamiliar to their historical predecessors I would guess that armored cavalry wouldn't have been abandoned until the 19th century and the stop start Tanzimat reforms.

My source for this being, as far as I can remember, Imber's Ottoman Empire (chapters 8,9), G. Agoston's work generally would interest you but specifically Guns for the Sultan, Murphy's Ottoman warfare, Ágoston, “Firearms and Military Adaptation”, and a couple others I could dredge up. These people are generally on academia.edu (especially Agoston) or JSTOR if you have access to either of those.

That said until the late 17th century (or later) the tartars of Crimea (read: Mongols) were the vassals of the Ottomans and would raid in the traditional manner frequently. Furthermore, the decisive charge that won Vienna in 1683 was by 20 thousand Polish lancers -- this questions is very much not cut and dry.

David Nicolle (not a historian by trade) has an excellent overview series on many of the main Islamic fighting groups from the early Islamic to Early modern Periods and are a great resource if you're interested in these sorts of questions generally.