Strategy for connecting 2 points without intersecting previously drawn segments of curves I have to connect point pairs without intersection. 
Let's say I have two given points that I connect with a segment of a curve. Then again two new endpoints are selected and these new points have to be connected as well however without intersecting previously drawn curves and so on for any number of given point pairs.

What is the easiest way of finding and drawing these segments of curves?
The general problem is called graph drawing and is not an easy problem.
The graphs that can be drawn on the plane without crossings are called planar, but not all graphs are planar: the typical graphs that are not planar are the complete graph on $5$ vertices $K_5$ and the complete bipartite graph on 6 vertices $K_{3,3}$, famous because of the three utilities problem.
You may want to try Graphviz - Graph Visualization Software to produce nice drawings.
See also

Szirmay-Kalos, László, Dynamic layout algorithm to display general graphs, in Graphics Gems IV, 1994. code
Rosati, Claudio, A simple connection algorithm for 2-d drawing, Graphics Gems III, 1992, code.