For box-box collisions (assuming you're not talking about axis aligned boxes, since those are easy), look up the separating axis test. It's a general test that works for any pair of convex polyhedra.
Box-sphere is a little bit more involved. The general idea is to expand the box by moving each face of the box outward by the sphere's radius, then test if the center of the sphere (a point) is inside the expanded box. However, for the test to be exact the expanded box should not be a sharp-edged box but a rounded box, with cylindrical edges and spherical corners. You can treat the expanded rounded box as being made of 3 regular boxes, 12 cylinders, and 8 spheres, and test if the original sphere's center is inside any of these.
Regarding the quote from the last article: imagine a very thin wall and a small, very fast moving object. In one frame, the object might be entirely on one side of the wall, and in the next frame, the object's moved far enough that it's entirely on the other side of the wall. It *should* collide with the wall, but this isn't detected because the collision occurs in between two frames. So to detect this, you need to test whether the wall collides with the entire volume of space that the moving object occupied in the time from one frame to the next. This is called a sweep test, and the article tells how to do sweep tests for a variety of shapes.
However, then in the case of a fast moving object, you want to determine the exact time that the collision occurred so you can apply physics properly. The sweep test lets you efficiently narrow down the range of time when the collision could have occurred, using a binary-search-like algorithm to find the moment of collision. (I don't think it's necessary to do as the article suggests and determine both lower and upper bounds for the time of overlap and then feed them to a different algorithm.)