Fractional Frequency and negative Frequencies
On figures 1, 2 and 3, you are plotting 2 curves which you call real and imaginary parts. These can be interpreted as the complex exponential

$$
x[n] = e^{jn\theta_0}
$$

with $\theta_0$ being 0, $2\pi/8$ and $2\pi/2$. These are not cosines, but complex exponentials.

Now, on figure 4, you only draw the sine part (what you call Imaginary), and not the cosine part. So this is why you don't find a cosine part!! You forgot to draw it. It has nothing to do with aliasing (which you interpret correctly).