Unless I'm missing something obvious, the answer to the original question is:
To have only three regions, the two lines cannot intersect inside the circle, so the each line must independently divide the area into 1/3 and 2/3 regions.
But if one point is at the centre of the circle, any line through it bisects the circle, so cannot produce the required 1/3 region.
And the answer to the bonus question is:
Any adjacent pair of regions must sum to 1/2 the total area, and are separated by one of the two lines. From this it follows that both lines must bisect the circle.
All lines that bisect a circle pass through its centre, so both lines must do so. And to divide the circle into four equal regions, they must be perpendicular.
This will not be possible for most pairs of points, but for a specific counter-example, consider any pair of points on the circumference that are not 90 degrees apart.