1. the phase plane was describe in $x-\dot{x}$ plot. and the nonlinear system was described in $\dot{x}=f(x)$. Phase plane could looks like this

This is telling us that this is not a function of $x-\dot{x}$. one x value signifies possibly two $\dot{x}$ (which actually IS) this is probably because the non-linearity make the same x could lead to different $\dot{x}$ such as $x^2$ (in which case the plot would be a sphere)

1. Banach Fixed points theorem is a sufficient condition. which means, this is a stronger condition to have a fixed point, but not nessesarily so. try draw some plot and you would find: contraction mappings always have fixed point, some contraction plus expansion also does, but if you plot it all above $y=x$ then this is too much expanding. That case no fixed point after iteration