Why did you expand around $x=0$ and how did you know which expansion to use?

I actually expanded around $x_0$. The Taylor expansion is generally valid, about any point. You can always write $U(x) = U_0 + x\left(\frac{dU}{dx}\right)_{0} + \frac{x^2}{2!}\left(\frac{d^2U}{dx^2}\right)_0 + ...$ about $x=x_0$, where the derivatives are evaluated at $x=x_0$.

This is a useful technique to understand the behavior of the function around some point, at small distances away from that point. We did this today for $U(x)$ around an equilibrium point, and showed that the second-order term $\frac{x^2}{2} \left( \frac{d^2 U}{dx^2}\right)_0$ is the important one to describe motion near the equilibrium point.