How long for the sun to traverse its diameter at the pole? As the sun rises once per year at the north/south pole, the time taken to rise (and set) must also take significantly more time to complete to rise (and set).

How long does this take at either pole? 

Interested to know if it is the same percentage of a day as anywhere else on earth (the day at the pole being several months long).
The Sun's daily motion does not affect its altitude on the poles, because it revolves around the zenith (and the zenith coincides with the celestial pole). This means the Sun's altitude is only altered by its movement on the ecliptic. The ecliptic is tilted at a roughly 23.5 degree angle to the equator (which is the horizon if the observer is on the North pole), and only the vertical component of its yearly motion contributes to its increase in altitude. Therefore, in order to estimate how long the Sun takes to rise/set on the North pole, let us assume the Sun is moving across the ecliptic at a constant velocity and that Euclidean geometry holds for the small area on the celestial sphere around the Sun. If the Sun takes one tropical year to traverse 360 degrees, its "speed" will be $\rho=0.98565 \frac{\circ}{d}$, and the vertical component $\rho\sin(\epsilon)=0.39\frac{\circ}{d}$. The time it takes to traverse a solar diameter is (roughly $0.5^\circ$) is $0.5/0.39 \sim 1.25$ days. The same is for the South pole, everything is rotated around the equator. You can find more details about the topic here.
As for the ratio of sunrise duration to daytime duration, these are affected by different phenomena, and the ratio of sunrise duration to daytime duration is not constant throughout the year and depends on the latitude of the observer so comparing them is quite ambiguous.