Even and odd signal energy property
Your result is correct but note that for complex signals, the even and odd parts are defined by

$$x_e(t)=\frac12\left[x(t)+x^*(-t)\right]\tag{1}$$
and
$$x_o(t)=\frac12\left[x(t)-x^*(-t)\right]\tag{2}$$

where $^*$ denotes complex conjugation.

From $(1)$ and $(2)$ it follows that the real part of $x_e(t)$ is even and its imaginary part is odd, whereas $x_o(t)$ has an odd real part and an even imaginary part.

We get

$$|x(t)|^2=|x_e(t)+x_o(t)|^2=|x_e(t)|^2+|x_o(t)|^2+2\,\text{Re}\{x_e(t)x_o^*(t)\}\tag{3}$$

From

$$\begin{align}x_e(t)x_o^*(t)&amp;=\frac14\left[x(t)+x^*(-t)\right]\left[x^*(t)-x(-t)\right]\\&amp;=\frac14\left[|x(t)|^2-x(t)x(-t)+x^*(t)x^*(-t)-|x(-t)|^2\right]\end{align}\tag{4}$$

we see that $x_e(t)x_o^*(t)$ is an odd signal according to definition $(2)$ (it is the odd part of $\frac12 \left[|x(t)|^2-x(t)x(-t)\right]$), and, consequently, its real part is odd. This implies that the integral over the last term in $(3)$ is zero, and the stated property holds.