There's parts of your question that I'll leave up to people far more educated than myself on the subject to answer, but I'll try to clear up some misconceptions.

Firstly, you state that "in 100 years it was all gone." That's not quite true: the entire eastern portion of the empire survived on the as the Byzantine Empire for almost another thousand years. However, the Western Empire rolled over and died completely. Also, Roman power returned to Italy during Justinian's reign, albeit briefly and weakly thanks to the plague and the Lombard invasion.

Secondly, the empire of 375, while appearing in terms of territory similar to that of Rome's golden age, was far from the days of Augustus. Plague, famine, and constant war during the Crisis of the Third Century had diminished the population of the West, already less urbanized than the eastern part of the Empire, quite substantially. Furthermore, the empire was suffering from attacks and settlement by Germanic peoples like the Goths around the late 4th century: far before end of the Western empire, these "barbarians" came to dominate much of the military and part of the government as well. So, in fact, the final Romans' grandparents wouldn't have been born into the great empire you might be misled by a map into believing still existed. Most of those outer regions soon became extensively settled by "barbarians" like the Goths, Franks, and Burgundians, and were semi-autonomous if not independent in all but name anyway.

That being said, the Western Empire wasn't in an un-salvageable state at the beginning of the fourth century. When the province of Africa, a vital, rich, and mostly undisturbed region fell to the Vandals, everything really started to fall apart.

Finally, I'll give a stab at you asking what people formerly part of the empire would have thought. I'll say, first of all, that it's difficult to determine what the common man would have thought: it's not like he had the ability or the will to express himself. However, the "barbarian" rule in certain parts of the empire, such as Italy, really wasn't that bad and not too much of a disruption from Roman rule. Rulers like Theodoric consciously tried to preserve Roman traditions and were respectful of the native culture. In Iberia, the formerly Roman populace had tensions with the Visigoths, who followed Arian Christianity as opposed the the mainstream Chalcedonian beliefs. In Carthage I know that the Vandals had some mildly prosperous trade and that the region was still rich enough to be Justinian's most valuable conquest, indicating the people were probably still doing alright for themselves.

Sources:

A History of the Byzantine State and Society, Warren Treadgold

Byzantium, The Early Centuries, John Norwich

History of the Decline and Fall of the Roman Empire, Edward Gibbon. (has more than a few issues...)

I wish I could answer certain parts of your question more in-depth, but my specialty in is Byzantine history, which often lightly sweeps over the events in the Western Empire during late Antiquity. I think your questions are pretty good ones, though it's difficult for anyone to answer some of them. I'm looking forward to seeing one of our resident Roman experts try to take a crack at doing so!

On the subject of Gothic Italy and Vandal Africa: in both cases these states can look from our perspective as post-Roman, i.e. whatever rose up inside the ruins of a fallen empire. As per your question, we might expect that in both cases there might be at least some nostalgia or regret for the empire that had gone. But in both cases there seems to have been a strong theme of Roman-ness and continuity with the forms Roman rule that gave legitimacy and stability to both regimes. James O'Donnell's Ruin of the Roman Empire, a book I really think quite good, advances this Roman continuity as the key to getting why these two states flourished (and why they proved such trouble for Justinian). Theoderic ruled very much as the emperors before him had done, and in the case of his immediate predecessors, his version of roman rule was much more effective. The Vandal case, although less well sourced than Theoderic, has similar outlines. In both cases when Justinian's armies arrived in the sixth century, ostensibly to bring these "Roman" places back into the empire to which they rightly belonged, the local populations treated Justinian's forces more like foreign invaders than fellow-countrymen come to rescue them from the barbarians. In Italy this stark resistance to the empire on the part of the "Roman" inhabitants of Italy explains why the Gothic war dragged on so long. In North Africa the Vandal regime fell quite quickly, but low level (guerrilla, to use the wrong term) resistance ensured that Constantinople would not have an easy time holding on to this conquest. I know that O'Donnell's book is controversial on these points, but if he's correct, or close to it, then the implication for your question seems to be that view the empire as something sadly lost, the citizens of the western successor states were much more likely to see Roman-ness as ongoing and the emperor in Constantinople as a dangerous foreign aggressor.