In chat, I was informed that $\operatorname{Im}()$ and $\operatorname{Re}()$ are homogeneous functions of degree $1$, as well as linear.
The issue I'm having is moving imaginary constants in and out of these functions. For example, $3i\operatorname{Im}(e^{ti}) = 3i\sin(t)$, while $\operatorname{Im}(3ie^{ti}) = 3\cos(t)$, but $3i\operatorname{Im}(e^{ti})$ and $\operatorname{Im}(3ie^{ti})$ should be equal by homogeneity and linearity.
Are these functions really homogeneous and linear? Do such notions work abnormally when scaling by imaginary factors? Otherwise, what else is going wrong here?
There are a few different definitions of "linear".
The functions $\Re$ and $\Im$ are "$\mathbb{R}$-linear"—that is, $\Re{(kx)} = k\Re{(x)}$, and $\Im{(kx)} = k\Im{(x)}$, for any $k \in \mathbb{R}$.
However, they are not "$\mathbb{C}$-linear"—that is, those equations don't necessarily hold for $k \in \mathbb{C}$.
Notably, these definitions don't require $x$ and $k$ to belong to the same space. This part of linearity is all about $k$, the factor that you can "pull out", and what sort of thing $x$ is doesn't matter. In particular, linearity is often used in vector spaces, where $x$ is a vector, but $k$ is a scalar. And if not otherwise specified, scalars are generally assumed to be reals.
(For the sake of completeness, as you probably already know, linearity also requires $\Re{(x+y)} = \Re{(x)} + \Re{(y)}$. But that doesn't have anything to do with scalars and isn't affected by what scalar field you choose.)