I can deal with a few of the questions in detail, but I'm afraid you have to wait for others to answer the rest of them!

What do Byzantine sources have to say about the Persians generally? Did Byzantine and Persian royalty ever intermarry, before and after the Muslim Conquest? What were Byzantine/Persian relations like after the Muslim Conquest of Persia?

I should preface this by saying that our source material for this is so very bad - we have no Persian historical sources and no Greek historical sources between c.630 and the ninth century, so unlike the sixth century we don't have any detailed contemporary accounts of Roman-Persian relations. I will also use Roman here throughout my comment to refer to the Byzantine Empire, since I don't see the point of distinguishing between the two, particularly in late antiquity.

Firstly, my answer on the Roman-Persian wars of this period will provide a useful introduction, since the last war of 603-628 directly preceded the Arab conquests and set the scene for domestic troubles in both empires even as the Arabs marched from Arabia (my answer on the reasons behind the Arab conquests' success is less relevant, but should still give you some idea of what this period was like). Generally speaking, Roman authors in the sixth and seventh century acknowledged the Persians' power and prestige, as well utilising widely Persian sources (both anecdotal accounts and official sources) in their histories. However, the same historians also described the Persians as decadent and duplicitous, which frankly wasn't that unexpected - Greek/Roman historians had a long and proud history of belittling anyone who wasn't a Roman. More officially however the Roman state maintained a cordial diplomatic relationship with the Persians when they were at peace, as trade between the two was quite important and the other empire's co-operation was pretty useful in controlling their respective client kingdoms in the borderlands between the two powers. This was true even in war-time, since the two empires developed a complex diplomatic ritual with each other, which can be seen in the detailed guidelines concerning how Persian embassies should be welcomed/treated preserved in the tenth-century Book of Ceremonies (based on the sixth-century work of Peter the Patrician, an important imperial diplomat). We can only assume that the same was true for Persians welcoming Roman peace embassies, even though we don't have any sources for that.

Conflict was however an important part of Roman-Persian relations in this period and this can be best seen in the last war between the great powers, which saw the seizure of all of the Roman Near East (an unprecedented military success for the Persians), the siege of Constantinople in 626 (the first siege of the capital since its foundation) and the realignment of Roman propaganda/imperial rhetoric to counter this threat (in some people's perspectives almost akin to calling for a holy war against Persia). This was however not the norm and it is clear that imperial propaganda does not provide us with a full picture. Between 591 and 602, the two empires were actually co-operating on a massive scale. This was because Khusro II was only returned to his throne with the help provided by emperor Maurice. As a result, Khusro gave Maurice massive territorial concessions and the two co-operated extensively in matters of faith and politics. Khusro may also have married one of Maurice's daughters, but it is I think quite unlikely. With Maurice's murder in 602 and his eldest son Theodosius' flight to Persia, Khusro had ample justification to wage war against the usurper Phocas in order to avenge his benefactor, though obviously he also had many more cynical reasons to intervene - the Romans were internally divided, so why not try to take back some of the territories he had lost a decade earlier?

This war naturally made many Romans very angry. The contemporary Paschal Chronicle for example consistently called the Persians 'accursed' and many other sources deplored the Persians' treatment of the Romans, with one angry monk later recounting that the Persians had forced Christians to trample on the True Cross. The reality was however I think much more tame. With the fall of Jerusalem in 614 and Alexandria in 619, the Persians realised that they can actually rule the Near East and began to consolidate their hold on these provinces. Jerusalem is the best example, as it was first surrendered peacefully and was only sacked when the city revolted against Persian rule; even then, the revolt seems to have been precipitated by Christian anger at how the Persians had favoured the local Jews over them. This resulted in the brutal sack of the city and the exile of many Romans, including the patriarch of Jerusalem, along with relics such as the True Cross. This was an event 'worthy of unceasing lamentation' (words of the Paschal Chronicle), but I don't think it reflected normal Persian policy during the war. The Persians certainly favoured Jews and miaphysite Christians (heretics in the eyes of Constantinople, which generally followed the Chalcedonian brand of Christianity) as part of their divide-and-rule strategy, but soon after 614 the local Chalcedonian hierarchy in Jerusalem was once again influential and began to rebuild the city. Jewish influence was also curtailed at the same time. We have to remember that the Persian Empire also had a big Christian minority in its heartlands, so the Persians could not treat local Romans like an underclass forever. Instead they sought reconciliation and began to rule the locals as subjects, not as sources of loot to be taken away back to Persia.

This spirit was also adopted by the Romans, an attitude that we can only begin to discern after their miraculous victory in 628 (though perhaps also before then, we have a Roman letter from during the war in which they literally begged for peace - for all their rhetoric, the Romans did not and more importantly could not ignore the Persians' ascendancy in the 610s). The emperor Heraclius did not demand too much from the Persians in the resulting peace treaty and had even offered to help a Persian general, the 'accursed Shahrbaraz', to seize the Persian throne. Although Shahrbaraz soon died after seizing power in 630, in the interim he was able to do a lot - he returned many Christian relics to Constantinople (like the True Cross, the Holy Sponge and the Holy Lance) and two of his children became Christian patricians, with his daughter even marrying one of Heraclius' sons. This son was actually a deaf-mute, which suggests that this marriage was very much a demonstration of the close ties between Heraclius and Shahrbaraz rather than anything else. Indeed, the last Greek historian of this period, Theophylact Simocatta, described the relationship between the two empires in these terms:

God effected that the whole world should be illumined from the very beginning by two eyes, namely by the most powerful kingdom of the Romans and by the most prudent scepter of the Persian State. For by these greatest powers the disobedient and bellicose tribes are winnowed and man’s course is continually regulated and guided.

Admittedly he was describing events in the 590s and he may have only used the rhetoric common in that more peaceful time, but I think it is an accurate enough depiction of the attitude in Constantinople c.630. Peace had finally been achieved and there was no appetite for further war, so a return to the status quo of the sixth century was perhaps desired by many. We of course know that the Arab conquests would soon destroy all that, initiating a new 'world crisis' and a 'Mediterranean world war'. I can't really talk about events after the conquest and as I said above the source material for the conquest itself is slim. As the Persian Empire had been shattered by the 650s, I assume you want to know about Romans' relationship with the Persians who now lived under the caliphate, which I unfortunately know nothing about. However, if this extract from Matthew Canepa (a historian of both Rome and Persia) is any indication, the resulting interchange of ideas was pretty interesting, so I look forward to any answers from people more knowledgeable about this period:

With the collapse of the Sasanian empire and the radical transformation of the Roman empire after the Muslim invasions, an elite inhabitant of late seventh-century Constantinople or Ctesiphon would have certainly experienced the world as he knew it dissolving before his eyes. Yet the traditions of the earth’s two eyes survived this cataclysmic period and endured long after the empires themselves fell. After the dissolution of the Sasanian empire, members of the court fled to Constantinople and Tang China, establishing two centers of resistance and agitation for the resurrection of the empire. Although unable to reconstitute the empire, their efforts ensured that Iranian culture continued to be felt in both the East and the West. [...]

Despite the destruction of the old order, Rome and Sasanian Iran’s relationship indelibly imprinted world politics and symbolics of power for almost a millennium thereafter, serving as a prestigious template for the exercise of power and diplomacy from the Atlantic to Central and South Asia.

The normans first came as mercaneries for the lombards lords of southern italy. De Hauteville had like 12 sons from 3 different wives and he did't know what to do with them, so they were encouraged to get out and find a purpose in life. The first mercanery detachment got beaten by the Byzantines, but they proved to be well-abled fighters, so the lombards ordered more. As there were a lot of landless Norman knights and thirdborn, supply was easy.Over time the Normans established themselves as a good mercanery unit in Southern Italy and they were used by both the Byzantines and the Lombards. As they were sometimes paid with land instead of money, they begun to form their own Kingdown.

Source: Frank Westenfelder, 2011, Eine kleine Geschichte der Söldner: Historische Gestalten auf dem Weg in die Moderne which is a German book about mercaneries.

To what extent was the Western European tradition of chivalry adopted by the Byzantine military-aristocratic class?

While you are waiting for more thorough answer. I feel confident enough to maybe answer this single question.

In short, no, or at least not in the West European style which was developed around 12th and 13th century. In Byzantium, this was also time of the infamous Fourth Crusade that sacked Constantinople so eastern Romans probably didn't get a very good impression of what western code of conduct would be. In fact, even during earlier crusades when Byzantium helped to deliver crusaders into the Holy Land, they were considered by Romans to be rather oafish (to put in mildy) and played their role of foreign barbarians, which Romans still considered them to be, rather convincingly.

Also, Chivalry in Byzantium wasn't really necessary. Chivalry was heavily encouraged by Catholic Church in order to keep local warlords from infighting (or at least to do it more safely which is where tournaments came from) and to protect their own property from said warlords. Byzantium didn't really have this problem or at least had more pressing problems to deal with at that time.

My sources for this would be The High Middle Ages and The Late Middle Ages by The Great Courses, Lost to the West: The Forgotten Byzantine Empire That Rescued Western Civilization by Lars Brownworth and overall discussion on this sub.