Step 1 :First, we analyze the function $f(x) = -\sqrt[3]{x}$.
Step 2 :Since the cube root function is always increasing, the function $f(x)$ is always decreasing.
Step 3 :Since the cube root function is defined for all real numbers, the domain of $f(x)$ is all real numbers.
Step 4 :The range of the cube root function is all real numbers, so the range of $f(x)$ is also all real numbers.
Step 5 :Since the function $f(x)$ is the negative of the cube root function, it is a reflection of $y = \sqrt[3]{x}$ across the x-axis.
Step 6 :Finally, we check if the function passes through the point $(3, -27)$. We have $f(3) = -\sqrt[3]{3} = -27$, so the function does pass through the point $(3, -27)$.
Step 7 :Thus, the correct statements are: \(\boxed{\text{The function is always decreasing, the domain is all real numbers, and the function is a reflection of } y = \sqrt[3]{x} \text{.}}\)