How does the curvature of space-time explain a ball falling in reverse? If we throw a perfect uniform ball up in the air without any rotation in a vacuum room in a perfect perpendicular direction of earths gravitational pull and mark the part of the ball that is pointing towards the upward/away direction with an arrow, then I assume that the ball at some point will stop moving away from
the earth and then start moving back towards the earth until it hits the earth.
In this experiment I assume that the ball will fall still having the arrow pointing
away from the earth.
How does curved space-time explain this? The intuition you get after reading a bit on the subject (not delving into the math) is that the ball should follow some path through space(-time) which is then being curved by earth. In this logic, it seems like the ball should carry on going "forward" along its path and eventually hit earth with the arrow poiting "forward" i.e. straigt towards earth.
But the ball stops at its peak in the air and then fall in reverse.
Curious how this works withing the curvatures of space-time.
And a similar quesion: If a photon is fired from inside a black hole in a perfect perpendicular angle away from the center of the black hole, how does the photon end up hitting the center? Like the ball above? If we mark the end of the photon initially going away from the black hole, is it this end that hits the center or the other end of the photon?
 A: 
In this logic, it seems like the ball should carry on going "forward" along its path and eventually hit earth with the arrow poiting "forward" i.e. straight towards earth.

"Forward" in the sense of a space-time trajectory means "future-directed." The path followed by the ball (or rather, each point on the ball) is indeed future directed as per this spacetime diagram, in which the direction of the worldlines is provided by the purple arrows and $z$ is the vertical coordinate.

The dotted lines appear to be curved, but the thing to understand is that the underlying spacetime is what's actually curved; the lines are as straight as lines can be when drawn on a curved surface.  Indeed, one has to be careful to define what it even means for a curve to be straight when the surface it's drawn on isn't a flat piece of paper; how can you draw a straight line on an egg, for example? This technical redefintion of straightness, along with a careful definition of intrinsic curvature, are laid out in the mathematics of general relativity; it is in this sense that (a) spacetime can be curved, and (b) the worldlines of inertial point particles are "straight."

If a photon is fired from inside a black hole in a perfect perpendicular angle away from the center of the black hole, how does the photon end up hitting the center?

There's no way to make this intuitive, but the clearest explanation I know is that at the event horizon, the radial coordinate and the temporal coordinate switch roles.  What this means is that within the event horizon, the phrase "if a photon is fired away from the center [...]" takes on the same meaning as the phrase "if a photon is fired into the past [...]" does outside the event horizon.  Once this role-reversal occurs, there simply are no light-like trajectories which proceed away from the center.
A: You are picturing the motion as if it only occurs in space and then having difficulty understanding the curved nature of spacetime.  Within space, the top of the ball will stay in the same orientation, as long as - as you specified - the ball is not rotating.  You must add in a time dimension in order to understand the curvature.  As in other answers, this is not a perfectly drawn example, but gives you a rough idea.

Within in space (left side above) the ball simple goes straight up and straight back down.  Within spacetime (right side) you see that the ball travels along a curve.  But note that this does not change the orientation of the top of the ball or the bottom of the ball.
A: I'm a little unclear on exactly what's troubling you.  The picture below is an approximation of the ball's path in spacetime.  (It's imperfect because it should be a parabola, but it was easier for me to draw something more like a semicircle.)  Imagine drawing this on a rubbery piece of paper.  Are you able to imagine stretching and distorting the paper in a way that converts this path to a straight line?

A: Your question assumes that the ball always points in the direction in which it is moving through spacetime. There is no reason to suppose that would be the case. If you walk up and down a hillside your body remains vertical (ie your head above your feet)- the orientation of your body does not change.
