Quote by CrossboneSRB
So here's a question for you, if on a line we chose one point, then what is the fraction of this point compared to the line?

0. The "length" of a single point is 0.
Quote by CrossboneSRB
I think we all agree that between two points there are infinite amount of other points, correct? So what fraction of a line is one point among infinite others? Isn't it 1/∞?

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.
Quote by CrossboneSRB
Here's another question, if a perfectly round object is placed on perfectly flat surface, what is the area of contact?

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.