Drift current and reverse saturation current
Diffusion will happen at any voltage. It is only dependent on the concentration of electrons or holes. However, the diffused electrons and holes will give rise to an electric field, causing an opposite drift current. So you end up with a balance with both diffusion current and drift current in opposite directions.

Even in reverse bias, you will never be able to completely remove diffusion current inside the junction. However, the global current through the diode is only concerned about which electrons and holes make it through the ends of the PN junction, not just inside the junction.

The formula for diffusion current is

\$J_{local}(x)=-q\cdot D\cdot \frac{d\rho(x)}{dx}\$

Where \$J_{local}(x)\$ is the current density at x, and \$\rho(x)\$ is the charge concentration at x. \$D\$ is the diffusion constant and is dependent on the temperature!

In 3 dimensions:

\$\vec{J_{local}}(\vec{r})=-q\cdot D\cdot \vec\nabla\rho(\vec{r})\$

So no matter what happens, as long as you have more electrons on one side, and less on the other side, there will be non-zero diffusion current.

[EDIT]
To answer your second question:
Reverse bias current is caused by minority carriers that are thermally generated (eg. electrons in a P-region). These carriers will randomly move just because of their thermal energy. This is in fact what diffusion current is for a single electron.

These minority carriers can be caught by the electric field in the depletion region, causing them to move due to drift to the other side (eg. our P-region electron will drift towards the N-region).

If there was no drift, then the net current generated by all the moving minority carriers (diffusion current) would average out to 0, as some would move left and some would move right. No global current would flow. So technically, you only have a reverse bias current because of drift...