The world has Fibonacci-like properties all over, not because we choose to view it through that lens, but because it's such a simple and ubiquitous quality of reality. In many cases it's easy to see how "the new value is the sum of the two previous values" is a concept that describes basic processes. You would be hard pressed to find a more direct description of those effects.
Exactly.
The reason Newtonian physics is so successful is because almost anything can be described by a second order differential equation. The world behaves as simple as that and it would be quite weird if it required a larger order to describe it.
Yes but that works for this equation only because its characteristic polynomial is x²-x-1, whose roots are the golden ratio and its conjugate. For a general recurrence equation (order notwithstanding) you can get arbitrary asymptotics (dominated by the characteristic root of highest magnitude).
What I find interesting is that this class of sequence's ratio converges to the golden ratio as n approaches infinity, but the quantum system produced the golden ratio between its first two values.
This seems to suggest a more straight forward derivation (x + 1 = x^2) than infinite recurrence. [0]
If you want to solve a recurrence, you can do that by assuming the general form of the solution (which is c*a^n for the above), or transforming it into exactly that polynomial you listed, you'd get something similar by using generating functions too.
