Did Byzantines consider themselves Roman after the fall of Rome? I think I've heard that somewhere. Also, what pushed the steppe tribes to migrate west? I'm specifically interested in the Ostrogoths. What sort of tactics did the Ostrogoths use? How did romans feel about the migrators, both in Rome and in conquered areas, and did the conquered tribes view these migrators as liberators or simply new conquerors? And finally, for now, why was the Eastern Roman empire prospering, while the western empire collapsed?
Cheers in advance, any information will be really appreciated.
Yes, the Byzantines considered themselves as Romans as the Romans from Rome did. In large part Justinian's, as well as other Emperors', reconquest of the old Roman lands were seen precisely as that - reconquests of Roman lands by Romans. Likewise, the Romans in Rome considered the Romans in Anatolia and the ERE as Romans as well.
As for why the ERE prospered while the WRE thrived, I cannot speak in much depth as I'm not hugely knowledgeable of the era, but I can say for certain that economics played a large factor. This is because in large part economics also has effects on military and political matters. The WRE faced dangers from all sorts of different factors, not just economic ones, but the economic situation of the ERE allowed for it to enjoy far more stability and prosperity than it's Western counterpart.
The reason why the ERE had such a leg up on development is because it held some of the richest lands in the Empire. The Middle East, North Africa, and Anatolia had been the crucible of humanity and home to some of the most populated and well developed regions outside of China and India. Comparatively speaking, Europe was sparsely populated and comparatively undeveloped. This was further exacerbated by Roman investments in the ERE, especially after Constantine moved his capital to Constantinople.
You can see here what I mean by the ERE getting the better chunk of the empire. A large amount of Europe's total development existed in Italia, not the rest of the Western lands - and even the regions of North Africa that it got, essentially Tunisia and everything West of it, were the less developed of the African lands that the ERE got (the very well developed and prosperous Egyptian territories).