Is the current flow through the diode carried out by both holes and electrons?
Short version:

Yes, the current in a diode is carried out by both electrons and holes. Mainly electrons on the n side and mainly holes on the p side

Long version:

When an electron moves from the n side of the junction to the p side it will continue to travel on its way until it recombines with a hole. The recombination rate is a function of varying factoring including acceptor concentration and trap density. Likewise for a hole moving from the p side to the n side. 

Most diodes have fairly high dopant concentrations so these recombination rates will be quite high, so the electrons or holes won't make it too far into the p or n sides, respectively.

You can look at the "drift velocity" (\$\nu\$) of the electrons and holes in the semiconductor. Thats probably the closest thing to the "speed of the current flow". Drift velocity looks at the average speed of charge carriers, but there is a very wide range of velocities that make up this average.

The drift velocity is a function of carrier mobility \$\mu\$ and electric field \$E\$:

\$\nu = \mu E\$

While carrier mobility is a function mainly of temperature, total impurity concentration, and the semiconductor material in use. In most semiconductors the hole mobility is generally much lower than the electron mobility given the same impurity concentration. Therefore, it is generally safe to say that drift velocity in the p region is lower than that in the n region. Unless there is a very strong difference in dopant concentrations in the two regions.

The resistivity (\$\rho\$) of a semiconductor is a function of mobility as well:

\$\rho = \frac{1}{q(\mu_nn + \mu_pp)}\$

These values don't change based on the bias of the diode. However, it isn't really useful to look at resistance of the p and n regions in a vacuum since the main thing that controls the current is the barrier height.