Inclusion of points in rectangles using a sweepline algorithm
I think storing in the AVL-tree only bottom $y$ coordinate is not enough. I think you should store at least both bottom and top coordinates of the active rectangle. Since your rectangles are not overlapping storing only the bottom and top coordinates is enough to solve the problem.

Basically, storing active rectangle in the AVL tree needs to decide whether a point is in a rectangle. Consider the following scenario

  y1 ----------------------------
     |                          |
     |                          |
  y2 ----------------------------
         o y6
  y3 ----------------------------
     |                          |
     |           o y7           |
  y4 ----------------------------

  y5 ----------------------------
     |                          |
     |                          |
  y6 ----------------------------


Initially your $p$ points in the AVL tree. As you sweep you add both $y$ coordinates of the active rectangles in the tree (and remove them as you rich the end of the rectangle). 

If at the same time you have more than one active rectangles then by brute force method you would need to go over all rectangles and check whether $p_y \in [b,u]$. However, using AVL tree you could find the point $p_y$ in the tree and check the number of points smaller than $p_y$. If that number is even, say $2k$, then you have exactly $k$ active rectangles below $p$ meaning that $p$ is not in one of them, otherwise (if odd) it is in a rectangle. Look at the above figure, there are 4 points smaller than $y_6$ and so $y_6$ is outside, but there are 3 points smaller than $y_7$ and so $y_7$ is in a rectangle.