Firstly, a lot of this rehashes the "Speed and Spin off the Rubber" page, but this is more relevant to Part 2 in particular.
The three main things that affect power and spin (there are other factors) are:

1. Hardness. This obviously plays the main role in the power test, but the hardness (particularly of the topsheet) also plays a major role in the spin that the racket is able to produce. This is because the softer the rubber is, the surface area that contacts the ball is increased, thus there is more friction.

The elasticity is closely related to the hardness of the rubber. Many rubbers are stretched in the factory, and are sold as ‘tensor’ rubbers. These achieve more power by acting as a trampoline; the rubber compresses, and ‘springs’ back into shape, providing more power. Tensor rubbers have a similar effect on spin, much like when a spinning rubber ball bounces, and the spin reverses. The rubber of the ball stretches while contacting the ground, and springs back into place, reversing the spin. However, in the case of a table tennis rubber the tangential stretch of the rubber due to the spinning ball, coupled with the movement of the racket against the spin (increasing this stretch), the rubber returns to its natural state, imparting a great deal of spin on the ball.
 

1. Tackiness. The tackiness (or stickiness) can affect the power of the racket, as the bounce is reduced by the racket’s tendency to ‘hold on’ to the ball. Tackiness has a larger influence on spin than hardness of the rubber, as a slippery surface could have a greater contact area, but the ball would not grip. The tackiness provides a point for the ball to momentarily stick to and therefore be turned by the racket (known as topsheet spin), as opposed to the mechanical spin produced by the return to shape of the rubber.
2. Smoothness. Though smoothness does not affect power or spin as much as the hardness (and elasticity) or the tackiness (and grip), the smoothness plays a part in the contact area of the ball. If there is a large amount of dust, for example, on the bat, then not only would the dust cover grippy parts of the rubber, but the dust would also make the surface uneven, and if parts of the rubber are covered in dust, then that area is contacting the dust, not the racket, thus reducing the contact area of the rubber.

​Why the throw angle changes with spin:
When a spinning ball hits the racket, one side of the ball is travelling faster than the other. When the ball makes contact with the racket, the ball grips to the racket. This means a certain point of the ball stops, as the ball begins to turn away from the racket, as the ball naturally keeps spinning (due to inertia). Because this ball has gripped, the angle of reflection changes, becoming what is known as the throw angle.