While the lack of fuel was a serious concern to the Romans (even during the time of Augustus), I don’t think either the population pressures or the lack of fuel had much of an effect on the decline of the Empire.
After the 3rd century, Rome was more likely to face under population than overpopulation. It was estimated that the Crisis of the 3rd Century led to the deaths of about 1⁄3 of the Empire. Afterwards, (particularly in the West) the severe lack of people led to the development of the patronage system. That was one of the reasons why Rome couldn’t defeat Alaric in 410, they had no available people.
Also, for the most part, Rome was “full” even during the peak of the Empire. For the most empty provinces, Gaul and Hispania, they were empty not due to the lack of people, but due to the lack of arable land. The technology to start tilling the hard soils of northern Gaul wouldn’t be invented until the 10th century. So the late Empire wouldn’t have faced any more population pressures than the Empire of Claudius or Nero.
And finally, the lack of fuel was also a problem during the peak of the Empire. I’ve once heard (though I don’t have a source) that much of the German economy was selling fuel to the Romans circa 100CE. But besides that, the Romans had other alternatives to wood, such as how coal was often used by blacksmiths.
Oh, also I wouldn’t say that history has “moved” north overtime, just that the history of the predominant “western” empire has. If looking at the most powerful and/or sophisticated state overtime, the focus of history would constantly move between Egypt, Anatolia, Persia, Northern India, and Southern China for most of history. I think the main western empire has moved overtime simply because it can’t go south, west, or east (because of the Sahara, Atlantic, and because if it went East, it wouldn’t be considered western anymore).
I don’t necessarily agree with your depiction of the Romans as being “parasitic”. Just because they did not produce food, does not mean that they were not valued.
The Romans were interested in math, its just that most of them weren’t located in Italia. Just look at the various mathematicians who lived in Alexandria, Athens, or Constantinople, and invented the fields of trigonometry (among others).
Rome had almost completely absorbed Greek culture and academics, to the point where many prominent Romans often read and wrote in Greek. Unless you were Cato the Censor, you almost certainly learned Greek math, its just that if you wanted to practice it full time, you would live in the east (and spoke Greek). Especially after the 4th century, when the focus of the Empire shifted to the East anyways.
Also, the Romans heavily benefited the economy of the Greeks. An interconnected empire meant that Greek goods (such as amphorae, pottery, or other luxury items) could be traded anywhere in the empire, with only the nominal port taxes placed on it by the Empire. Also Rome wasn’t militarily occupying the East either, since the entirety of it was governed by the Senate (except Syria, Mesopotamia, and Armenia).