All the energy in a closed system is constant.

There's only one closed system in nature. That is the entire universe.


So if energy is "made"
in one area, it must have come from some other area in the system.

Only if your system is the entire universe. Otherwise it can easily come
from another system and, in experience, often does. People are still trying
to find the source of excess energy from the Pons-Flieschman experiments.



This
means that it is impossible to set up a perpetually moving mechanism
without an outside source of energy, as no motion can occur without
using energy.

This would violate the law of inertia. An object in motion stays in motion
unless acted upon by an outside force. An object moving along in free space
at any speed consumes no energy.

The law of inertia describes perpetual motion. In fact, the motion is
perpetual if no energy is added or removed from the system.

Where does the energy come from to induce additional rotation in the
Lens-Thirring effect?



Energy cannot be transferred from an area of lower energy to an area of
higher energy, without some external effect being felt in the system.

Is the effect external to/ outside of the system? If so, then it is not a
closed system and your sweeping macroscopic generalizations would not apply,
since they are valid only for closed systems.


So all systems must evolve from coherence to a state of randomness due
to random external effects caused by energy fluctuations.


Randomness is a mathematical concept used to describe or measure physical
systems. Randomness is not a force of nature, is not a physical process and
is not a cause for an effect. Randomness is used to describe aggregate
phenomena after it has occured.

In almost all systems coherence is the lowest energy state. Do some research
on oscillators.