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It the point is on the line the distance is 0. If the point is not on the line, then it is possible to draw a unique line from the point to the line which is perpendicular to the line. The distance from the point to the line is the distance along this perpendicular to the line.
Yes, a line and a point can be on the same line. A point can be placed on a line. It will then be collinear.
A point can be intersected by infinitely many lines. Two points intersect in only one line. Three points either intersect in a line or not at all. This is only considering two dimensions.
Yes because a line can lie in many planes so one we add one point not on that line, we define a unique plane.