This question has so many perspectives I'm not sure I can provide a reasonable or satisfactory answer to cover them. So, I'll focus on the general objective, with particular emphasis on providing "... a better sense of how Saladin's army looked and acted in the field*" (OP's comments).
Saladin's Military Mindset
The straight answer is Saladin's Ayyubid army adopted the method of Steppe Warfare.
This should not be surprising as he was, in his blood and upbringing, a nomadic warrior. Saladin was born in Central Iraq, Tikrit (1138), from the powerful Hadbānīya Kurd tribe, and steeped in the tradition of military families - he was nephew and son to 2 military governors of the Great Seljuk Sultan Muhammad Ibn Malik Shah.
Saladin grew up serving the Zengid dynasty - who were Orghuz Turks (Islamicised turk warriors) with direct military lineage back to the Seljuk Turks. Lest we forget, the greatest claim to fame of the Seljuk Turks was the Battle of Manzikert. 'Fame' because the Seljuk Turks were the first Muslim commanders who captured a Byzantine Emperor.
Whether Hattin, or Jerusalem, or the rest, Saladin's fights should not be seen as discrete events but rather a series of battles with an overarching objective, namely a holy war (jihad) in the form of a counter-Crusade (1187-1189). Without underestimating the importance of Jerusalem, Saladin's counter-Crusade was not just to unite Syria and Egypt. He was reclaiming Bilad al Sham - the original Islamic province of the Rashidun caliphate (the original Caliphate).
Remember, Saladin was a Sunni, whereas the Fatimids were Shi'ites. Because of Bilad al Sham, from the perspective of fellow Sunni Muslims, he was rebuilding Islam -- not just taking back castles/towns/cities. Also, that's why -- with his political base of Sunni Muslims strongly supporting him back in Egypt -- he never had to return home to pacify rebellions.
In this sense, his military legacy was beyond that of the Zengids - whether Nur al-Din (his overlord) or his father, Imad ad-Din Zengi. A better comparison is Khalid ibn al-Walid, the Rashidun Capliphate's first significant military leader (he united Arabia and won over a 100 battles in a span of 13 years, from 629 till his death in 642).
Hattin has been well-examined, especially by Western historians. There's more than enough material for research (not necessarily a good thing imho). If you're looking for more - though they may not be Saladin's army - but I would consider the following battles, given the geographical setting, just as useful in the in terms of 'look and feel' of the combat:
The Ayyubid Army
There was no significant difference between the Ayyubid army and the Franks in terms of military tools:
For a little under two centuries, from 1099 to 1291, Europe fought a series of wars and battles for control of the Holy places in the Middle East—the campaigns that today we call the crusades. Although the forces against which they fought over this period were many and varied, it is perhaps those of the great Muslim leader Saladin in the later twelfth century that had probably the greatest effect on the armies and tactics of Western Europe.
Much like the armies of Western European, those of Saladin were composed of both horse and foot soldiers in which the cavalry, though fewer in number, were again dominant—not only in the tactics and strategy adopted but also in the minds and hearts of those that fought—it is the mounted soldiers who were remembered both then and today. And again though very different in detail, the arms and armor worn by the Middle Eastern troops was broadly similar to those used in the West—the mail shirt, helmet, sword, lance, and bow, for example.
Medieval Weapons- An Illustrated History of Their Impact 2007, pp. 139-40.
In terms of ethnic composition, you have done most of the research (your listed biblio). If you're looking for recommendations on additional sources, I would add the following (the last 3 entries are also by Nicolle)
- Armies and Enemies of the Crusades 1096-1291
- From Saladin to the Mongols: the Ayyubids of Damascus, 1193-1260
- Arms & Armour of the Crusading Era 1050-1350 - Western Europe and the
Crusader States, vol 1 (Greenhill, 1999)
- Arms & Armour of the Crusading Era, 1050-1350- Islam, Eastern Europe
and Asia, vol 2 (Greenhill, 1999)
- Horse Armour in the Medieval Islamic Middle East (an academic
Combat Tactics of Heavy and Light Cavalry
Going through research on Hattin and the additional battles listed above, I believe you would discern a clear pattern in the modus operandi. Instead of just calling it nomadic hit-and-run for their light cavalry, I believe a better description is tactical swarming:
Many examples of military swarming at the tactical level come from the ancient world and the Middle Ages. The most common swarmer in history has been the horse archer, which was introduced into warfare by the nomadic barbarians of Central Asia. Swarmer-versus-non-swarmer battles usually involved light cavalry armies of nomadic people fighting infantry armies from more-settled agricultural communities. The Eurasian steppe produced most of the well-known mounted archers, including the Scythians, Parthians, Huns, Avars, Bulgars, Magyars, Turks, Mongols, and Cossacks.
The firepower and mobility advantages of the steppe warrior were not surpassed until the invention of gunpowder. Whether their opponent was Persian, Macedonian, Roman, Frank, or Arab, mounted archers usually fared well. Unfortunately, many of the ancient examples of swarming offer little detail because of the remoteness of the events and the lack of accurate and complete accounts. There are few ancient or medieval historical sources on the history of warfare between swarmers, because most swarmer armies were nomadic. Often, only a brief description of the conflict is available.
Swarming on the Battlefield - Past, Present, and Future, 2000 - p.13
For heavy cavalry, it is the standard close-quarter combat. I emphasise this because of the almost-overwhelming emphasis (by contemporary scholars) on light cavalry of nomadic warriors. They had heavy cavalry too. In Saladin's case, it would be the corps of Mamluks in the Ayyubid army, the Salahiyya - elite royal guard with the best training and armour. They were not just his personal bodyguards (more research - Mamluk Studies Review, University of Chicago)