I notice that when I plot the equation $\psi(x) = \exp(i k x)$, it forms a circle.
But right after, in the book I'm reading, it uses a wave function like $\psi(x) = \exp(i k x) + \exp(-i k x)$. This flattens the circle, squashing and stretching it.
Is it supposed to be like this, with flattened circles?
From what I can tell you are plotting the real component of $\phi$ on one axis and the imaginary component on another.
Now, by Euler's formula $$e^{ikx} + e^{-ikx} = \cos(kx)+\mathrm{i}\sin(kx) + \cos(-kx)+\mathrm{i}\sin(-kx) = 2\cos(kx)$$ (because $\cos(a) = \cos(-a)$ and $\sin(a)=-\sin(a)$). So you have no imaginary component and the function will be confined to the real axis, just as your images show.
In general I would expect to see coefficients on the terms in a wave equation. e.g. $Ae^{ikx} + Be^{-ikx}$, and then you would not necessarily end up with a straight line.
Also, a common thing to want to do is to plot $\phi^*\phi$, which is real and relates to the physical probability of observations, as a function of $k$ or $x$. That will usually not be a straight line over the range of interest.