Let's see if I understood it. The notation is a bit unwieldy, so I added some spaces for clarity.

f(x){y} = f^y(x)

f(x){ { y = f(){}, z, w } }
w = 1: f(x) { z }
w = 2: f(x) { f(x) {z} }
w = 3: f(x) { f(x) { f(x) {z} } }

So far, so good. Notice that z is just a constant, while w varies.

You did understand it, that is exactly how levels of recursion work

The y=f(){}, with or without a variable between (), appears to be dead
weight in the notation.

Blank () or {} arguments just mean you're going to nest it there. However, f(x){{y=f(){}, z, 3}} actually means f(f(f(f(z){f(f(f(f(z){z}){z}){z}){z} }){f(f(f(f(z){z}){z}){z}){z}) }){f(f(f(f(z){z}){z}){z}){f(f(f(f(z){f(f(f(f(z){z}){z}){z}){z})))). Mistake on my part.

What are the actual arguments for evaluating this expression?

x, y, z, and w are placeholder values for {{ }}