Your "feeling" is correct: Dynamical Systems Theory and Dynamics of Fluids will roughly cover your questions.
but which one covers most questions?
More relevant then the number of questions covered (which will anyway change as you study any subject) is what you consider the most important question? If your real interest is turbulence, than starting with fluid dynamics is the obvious path. If rather general dynamical systems interest you the most, then of course that's what you should study.
Minimum prerequisites for learning both these theories together are vector calculus and a good grasp of introductory undergraduate physics (check, e.g., Halliday, vol. 1), though you're not unlikely to need to learn at some point small bits of topology, thermodynamics, partial differential equations, statistical physics, number theory, etc. There are some general Book recommendations available, in particular: 1) Self-study book for dynamical systems theory? and 2) Book recommendations for fluid dynamics self study.
Some of the other more specific questions are a bit vague, possibly ill posed, but my take on them is:
Can a periodic event be described as a (hidden/imaginary) oscillator, even when its causes are still unclear?
Yes. You can describe periodic phenomena with the machinery of oscillators. It's perhaps ideal to obtain descriptions from known underlying physics, but that's not always possible, in which case we resort to phenomenological models (which is my interpretation of the "hidden/imaginary" bit).
When can't such system be approximated by simple harmonic oscillator?
Whenever nonlinearities aren't negligible. The simple pendulum is not approximated by the SHO for large angles. More examples and discussion: Why can some oscillations be modeled by Simple Harmonic Motion, while others cannot? and Why is the harmonic oscillator so important? .
can we theorize the periodic characteristic of seasons (spring-summer-autumn-winter-spring...) as an oscillator or a wave?
Yes. That's actually related to the fact that the projection (think "shadow") of a circular movement is a harmonic oscillation.
Is there a phenomenon that has an underlying dynamical process that is different to its surface (like a sea)? When that process goes up, how would it be?
I assume you mean whether it can happen that the underlying physics is not obvious to the observation: yes, certainly. And, by hypothesis, you can't tell in general what it'll turn out to be.
How does a turbulence spreads out in a media, and how to control it?
Lower Reynolds numbers are associated with laminar (i.e., non-turbulent) flows, so you can try to prevent turbulence from forming by reducing the Reynolds number of the flow. In general, we cannot control turbulence, but there are methods to achieve partial control. A review on the subject from an engineering perspective is Lumley and Blossey's CONTROL OF TURBULENCE.
