system function $H(\omega)$ relationship to odd and even components of h[n]
The DTFT relationships

$$x_{even}[n]=\frac12\left(x[n]+x^*[-n]\right)\Longleftrightarrow\textrm{Re}\left\{X(e^{j\omega})\right\}$$

and

$$x_{odd}[n]=\frac12\left(x[n]-x^*[-n]\right)\Longleftrightarrow j\,\textrm{Im}\left\{X(e^{j\omega})\right\}$$

hold for any sequence $x[n]$ for which the DTFT exists. There is no assumption about $x[n]$ being real-valued or causal (note the complex conjugation $^*$ in the definition of even and odd signals). If $x[n]$ is real-valued you can leave out the conjugation.

Note that the DTFT of the odd part $x_{odd}[n]$ equals $j$ times the imaginary part of the DTFT $X(e^{j\omega})$, so you have

$$X(e^{j\omega})=\textrm{DTFT}\{x_{even}[n]\}+\textrm{DTFT}\{x_{odd}[n]\}$$

(without a $j$ on the right-hand side).