Why does Coriolis determine the rotation direction of a cyclone but not my drain?
The answer is the scale.
The fluid movement of a sink has a much smaller curvature radius than the grand-scale movements of a hurricane.
This curvature radius plays a big role on whether your movement due to a pressure gradient will be balanced by coriolis, or centrifugal forces, as thorougly discussed here.

You can read this wikipage, but the essence is the following:

If you transform the equations that govern the movement of a fluid parcel into a coordinate system that moves along with the flow, you find the following force balance that counteracts the pressure gradient:
$$ \frac{v^2}{R} + f v = - \frac{1}{\rho} \nabla P $$

Here we have simply two effects that arise from the translation to a rotating reference frame balancing the pressure gradient. The first one is known as the centrifugal pseudoforce, the second one is the coriolis force.
If you're interested in a derivation, see here, the linearization is then straightforward.

From those terms one can clearly see that fast dynamics (tornado!), or small curvature (toilet) will lead to the l.h.s dominated by the centrifugal term. However for large-scale and slow flows (hurricanes, cyclones etc.) the left side will be dominated by the coriolis term.



As to the question in which direction a certain flow can occur, let's remember that in co-moving coordinates each scalar quantity has however a sign. $v$ changes sign if we turn the wind direction, $R$ the curvature radius sign depends on where the movement is going, $f$ on which hemisphere we are.

If we now take a look at coriolis vs. pressure gradient force (geostrophic balance), we see that this balance has only one direct solution for a sign of $v$ for a given f. Therefore the predetermined direction of geostrophic cyclones for each hemisphere.

However the solution for $v$ in the cyclostrophic case is obviously
$$v = \sqrt{-\frac{R}{\rho} \nabla P} $$

Which can never be satisfied for high-pressure systems as then curvature sign and gradient sign cancel out and leave the term imaginary.

However one can easily visualize where the forces are pointing:


($F_p$ is the pressure gradient force, $F_c$ Coriolis and $F_{ce}$ the centrifugal force. + and - denote the root's solution. Property of K. Roth, Heidelberg University.)

Here in the case of cyclostrophic balance just look where the red arrow can cancel the green one and you see how Tornadoes do not care for the hemisphere, but just for low-pressure systems and can go in any direction around this.