The Arab conquests quickly removed Egypt and Palestine from the Byzantine world. The Seljuk Turks moved into central Anatolia (this is what prompted the Crusades), and did seek to take Constantinople, although they were done in by the Mongols instead.
The Fourth Crusade resulted in the sack of Constantinople by Western Christendom. The Latin Empire didn't last terribly long, and the remnants of the Byzantine Empire took it back within a century, but the Byzantine Empire was at this point in a fatal cycle of internal rebellion. At this time, the Ottomans, who start out as a minor Beylik in western Anatolia, take advantage of Byzantine distraction to peel off the lands of Thrace and move their capital to Edirne and solidify themselves as the dominant power in the Balkans. It's from this base that they ultimately defeat the European powers (Battle of Varna) and put the Byzantine Empire out of its misery (conquest of Constantinople).
By and large, the Ottomans actually brought back stability to Thrace and Anatolia, at least for 300 years or so. Anatolia was lost to the Seljuks long before, and the Ottomans were the first power to be able to unify it under a polity since the collapse of the Sultanate of Rum. Christian infighting caused the political instability in the Balkans that the Ottomans were most able to take advantage of.
Note that the change in political control doesn't necessarily correspond to change in the general inclinations of the population. Greeks and Armenians still remained a major part of the Seljuk and Ottomans (although they did decline over the centuries), and the Ottomans were generally quite tolerant of this heritage for the first few centuries (much less so after around 1700).
A bit hyperbolic, but what comes to my mind when I hear empire praise is an older quote:
"And where they created a dessert, they call it peace"
I mean yes, you can maybe not judge them by our standards (on the other hand, I was just reading about syria and all the different groups fighting there for oil and gas .. so where are our standards actually), but still, I don't like to hear praise for a invading force, even though it might be strong enough to prevent war for some time for its new slaves... (again hyperbolic, I know and also that you probably didn't mean to "praise", just stating the facts)
But thanks for your good summary!
The Byzantines were effectively forced into a mode where they had to choose between internal and external security, which is a large factor of their terminal decline. By the Fourth Crusade, the Empire was pretty much in a state where the surrounding powers had more reason to wait to pick at its corpse than to help it survive.
You mentioned Siege of Vienna. Remember that it eventually failed and it culminated with Battle of Vienna when Ottomans were completely defeated and soon after driven out of Europe. I would have thought that had the battle happened outside Constantinople instead of Vienna the history would have been changed completely. Vast parts of Ottoman Empire would probably be part of EU now.
I would then look into Carthaginian wars, famous general Hannibal and his fate. This was a crucial period when Rome was competing with Carthage for the top spot and at times it looked quite dire for Rome as Carthage had an upper hand.
Next interesting period would be Julius Ceazar and especially what happened after his death when his generals were battling to inherit his empire which eventually led to the split into eastern and western Romans. This split later continued when Christianity became mainstream religion and lead to Byzantine Empire.
You can look into the sack of Constantinople and wonder what would happen if Western Christians didn’t sack and destroy Eastern Christians allowing Islam to take over the region.
And then we come to the German rise as German lands become the heart of the indirect continuation of the Roman Empire in the form of Holy Roman Empire.
I don't have anything more specific about later history, but the spice trade and the Orient Express are two (I think) fascinating pieces of history that tie Western Europe to North Africa and eastern Asia. The history of the Crusades is also integral to understand the mixing of these areas... Europe wouldn't have cared if they didn't think east Asia was "theirs" in some way.
The History of Byzantium podcast picks up right where the History of Rome left off. It's mostly about the east Roman empire, but also covers the all its neighbors.
They created some amazing architecture and culture in Southern Spain.
So either the Umayyads had no use for camels or Dr Green's research doesn't include records from the period.
I believe I've heard about their sporadic use in medieval Europe before, but certainly never in the context of humiliation of ones enemies.
This really supports the whole "reality/truth is stranger than fiction" trope, as we would probably dismiss a novel or movie set in medieval western Europe with scenes that depict people paraded around on camels as "unrealistic".