I can answer a small part of your question, specifically vis-a-vis how the Eastern Empire viewed the lost territories. In the immediate aftermath of the abdication of Romulus Augustulus, the last person to claim to be Emperor of the West (although the vast majority of western territory was already under the control various, mostly Germanic tribes), Odoacer was deposed by the Ostrogoths under Theodoric, who assumed control of Italy with the legal justification that he was ruling as a viceroy on behalf of the Emperor Zeno; Theodoric had already been given numerous titles (including Patricius and magister militum) by Zeno, so in this way, the Eastern Roman court could at least pretend that Italy was still under their indirect control. It should be noted, though, that this was a shallow read of the situation, since Theodoric had been invading Eastern Roman territory and threatening to attack Constantinople itself, so these titles and permission to take Italy should be seen as at least in large part an attempt to buy Theodoric off and divert his attention.

Centuries later, Justinian I reconquered widespread portions of the old Western Empire, especially Italy, a large chunk of North Africa, and parts of Spain. His justification (to the decree that he needed one) was that he was returning these areas to Roman control, so the Roman Emperors in Constantinople still clearly regarded these areas as part of their domain. Many of the reconquered territories remained part of the Empire for quite some time until the Arab-Byzantine Wars (Africa, for example, was under Roman control again from 590-698). Eastern Roman control of parts of Italy (e.g. the Exarchate of Ravenna) and influence on the region continued well into the Middle Ages; note that the Normans in the 11th century in Italy were taking Byzantine towns.

This goes for both people who would have been considered roman citizens immediately after the fall and the feudal states that would soon develop. Other than the ruins left behind, were people even aware Rome existed in a few generations? If so, did they see roman times as "The good old days" or as a weak, broken state they were glad to be rid of? Were they aware of its eastern continuation?

This is the best part of the first part of your question. We know that 'Roman' nobility continued in much of the Western Empire, even in far off Britain, at least for a while. As to whether they 'mourned' the death of Rome, there was certainly some shock by the sack in 410, but the Empire itself died a slow death after that, and the writing was on the wall for everyone by 476. Most of these men were living in the reality of the day and focusing on positioning themselves to survive and thrive during the change. These men would have been quite aware of the Roman Empire in the East, but most of them made a pretty seamless transition to rule under the various Germanic invaders. Most scholars today would suggest that life for the average nobility in Italy, Gaul, and Hispania would not have been that much different. The main difference was that avenues of power that had been traditionally available (in particular, service in the Imperial administration) for the nobility was now replaced by Church careers. So, for example, we have men like Saint Remigius, a Gallo-Roman noble of the late 5th/early 6th century, who took a leading role in the new administration and baptized the Frankish king Clovis. Indeed, the Church became essentially the sole purview of the Roman nobility. In Rome, Marseilles, and elsewhere, the landed nobility that had dominated local politics before continued to exist. Pope Gregory I for example was from a noble Roman family, and the early Papacy was dominated by such families.

And what was Byzantium's view of the west? Were they happy to not have to deal with it anymore? Preferring their eastern territory and greek culture over the ways of the old western rome? Did they view them as territories that were rightly theirs and would eventually return to the fold? Were there any attempts to retake the old empire and how did the locals feel about it? Thanks in advance for the answers!

There certainly were attempts to reassert Imperial authority, of which Justinian's campaigns in the 6th century are the most famous example. He retook Italy, Africa, and southern Spain, but over-reached. The 'Gothic War' in Italy saw the peninsula, which we can probably assume was still very prosperous, devastated. Indeed, Southern Italy was devastated to such an extent that there is some suggesting that the long-term disparity of wealth between South and North dates back to this time. As for the cultural aspects, laws during Justinian's time continued to be written first in Latin (of which the Code of Justinian is the most famous). This code is the basis for the 'Roman Law' that many jurisdictions in the West aspired to, demonstrating the continued influence the Eastern Empire had at this time. The transition to an actively Greek empire had to wait until the dreams of reclaiming the West died with the Islamic invasions, and it was a process that occurred over a few centuries; by the beginning of the 9th century, Greek appears to have been the primarily language of law.