Fast algorithm to find points on one side of hyperplane?
The keywords you may be looking for are simplex range searching and half-space range searching. See chapter 16 of Computational Geometry: Algorithms and Applications. If you are interested in general simplices (intersections of multiple half-planes), then there are two data structures in particular that might interest you: partition trees and cutting trees.

In the following paper describing partition trees, we have query time of $O(n^{1-1/d}(\log n)^{O(1)})$, preprocessing time of $O(n \log n)$, and space requirement of $O(n)$.

Matoušek, Jiří. "Efficient partition trees." Discrete &amp; Computational Geometry 8.1 (1992): 315-334.

In a cutting tree, you consider the dual case (each point in your input is represented by a line in the dual plane) and you tradeoff space for query time. You get $O(\log n)$ query time at a cost of $O(n^{2+\epsilon})$ space. 

However, if you really only care about a single half-plane in your query, you can do better. In The Power of Geometric Duality, Chazelle gives an optimal data structure for half-space range queries with $O(\log n + k)$ query time and $O(n)$ space.