Apply the formula for the circumference of a parallel of latitude.

The formula is $C_{lat} = 2\pi R \cos\alpha$, where $\alpha$ is the latitude.
Here, $\alpha = 50^\circ N$.

The circumference of the line of latitude $50^\circ N$ is $\boxed{\text{25733 km}}$ (rounded to the nearest kilometer).

The difference in longitude $\theta$ is $10^\circ + 50^\circ = 60^\circ$.

The formula is $D = \frac{\theta}{360^\circ} \times C_{lat}$, where $C_{lat}$ is the circumference calculated in part (i).

The shortest distance PQ along the line of latitude is $\boxed{\text{4289 km}}$ (rounded to the nearest kilometer).

Let the unknown latitude be $\beta$.
The circumference of a parallel of latitude is $C_{lat} = 2\pi R \cos\beta$.
The length of the equator is $C_{eq} = 2\pi R$.
We are given $C_{lat} = \frac{1}{4} C_{eq}$.

Divide both sides by $2\pi R$:

The latitude at which the circumference of the circle of latitude is equal to $\frac{1}{4}$ the length of the equator is $\boxed{75.5^\circ}$ (rounded to one decimal place).