Is that true that non-stationary processes only includes trends, cycles, random walks, and combinations of the three? If yes, has this statement been proofed? If not, what are the other forms of non-stationary?
No. There are an infinite number of types of non-stationarity which you can easily dream up. For example, changing variance, changing skewness, level shifts, seasonality in the 6th moment, etc.
Also, cycles can be stationary. Yule (1927) studied cycles in sunspots using stationary AR(2) models, for example.
Is that true that non-stationary processes only includes trends, cycles, random walks, and combinations of the three?
Unless you define one of the three types or combination broader than I do, this would, exclude for example:
Any superposition of a non-stationary process and a stationary process.
Any process with a chaotic driver. Such a driver cannot be decomposed into periodic processes.
If we are talking about discrete-time processes (as implied by cycle): Any process with a quasiperiodic driver.
Any transient dynamics, i.e., a process that settles on (or asymptotically converges to) some stationary process after a while, e.g., a driven pendulum that takes some time to synchronise with its driver.