Why is the orthogonal line segment intersection algorithm $O(n\log n+R)$ instead of $O(n\log n + Rn)$?
It's a bit hard to get through your text since a lot of information is missing. However I assume that you have the following misunderstanding. 

Let me ignore the constants in the big-O for this argument since this is irrelevant. For every horizontal segment $h$ you perform an action that takes $O\log n + R_h$ time, where $R_h$ are the intersections you output. Then the overall running time is
$$\sum_h \log n + R_h =  n\log n +R.$$

Simply speaking, you only output every crossing once. So the overall time for the output is $O(R)$.