0. The "length" of a single point is 0.
No, because we don't calculate length on the basis of how many points are in an interval. We calculate length by subtracting the position of the point on the left from the position of the point on the right. The length of an interval [a, b] between two points (assuming a < b) is b - a.
The ratio of the "length" of a point to the length of the line segment [a, b] is 0/(b - a) = 0.
In any case, division by ∞ is not allowed.
Your question isn't very precise. A circular disk is a perfectly round object, so the area of contact would be the area of the disk.
If by "perfectly round object" you mean a sphere, there is only one point of contact, so the area of contact would be zero for reasons similar to what I already gave.