The vertical profile of horizontal windspeed is often given by the so-called "log-law":

$$u(z)=(u^*/k)\ln(z/z_0),\ \mathrm{for}\ z>z_0,$$

which can be found here.

As can be seen from the equation, when $z=z_0$, $u(z)=0$. But what happens below $z_0$? If we continue to use the same equation, for $z<z_0$, $u(z)$ becomes negative and changes direction, which seems absurd.

The key text here is "for $z>z_0$". It's telling you that, while you can evaluate the equation for other values of $z$, outside of that range the equation is not a valid description of the physical system. The equation could be written piece-wise to be complete:

$u(z) = \begin{cases} (u_*/k) \ln(z/z_0)& z>z_0 \\ 0 & z\le z_0\end{cases}$

But this doesn't really add anything useful. In practice, the "log-law" is used to describe the wind profile over 10s of metres and values of $z_0$ range from 1 mm to 2 m, so values of $z$ are likely to be in the valid region. If you do need to make calculations that close to the surface (in the interfacial sublayer) then you'll need a different equation anyway.