When a ball hits the ground, it becomes flat. Even if you watch a golf ball land on a cart path in slow motion, you will see it flatten before returning to its original shape and taking flight again. Elasticity is an object's property to bounce back to its former shape after distortion. Most balls involved with sports have this to varying degrees -- a baseball will bounce back to about half the height that a basketball will when both are dropped from the same height, with all other variables the same -- but an egg (even a hard-boiled one) does not. A super ball has a lot more elasticity than a ping-pong ball, which is one reason why it bounces so high when you drop it.
Yes, gravity pulls the ball to the ground in the first place; however, gravity's effect on an object in motion is constant, even when the object is bouncing upward. The constant effect is that gravity pulls every object toward the earth at a steady 9.8 meter/sec^2. Whether current velocity is headed upward, that constant drag is taking a toll on the ball in flight. This is why second bounces are almost never higher than first ones, and subsequent bounces get lower and lower.
Balls don't bounce as well when the temperature goes down but it's not just true for balls. In ice hockey, the pucks are kept in a freezer so they don't bounce as much when they're in play. To test this, leave some golf balls in the freezer or refrigerator overnight, and compare how high they bounce to some other golf balls that are at room temperature. Kinetic energy, which involves particle motion, is less when temperature drops, which means that the particles move more slowly in reforming the ball's shape, and thus, the ball doesn't bounce as high.
Ping-pong balls don't bounce very well since they're composed of nothing more than a light empty shell. A super ball is about the same size but will tend to bounce five to six times as high as a ping-pong ball. It has a hard rubber core that, when it hits the ground, stretches on the molecular level. This coiling collects itself and then expends kinetic energy in the opposite direction, much like the springs on a trampoline. In the game of baseball, there was some controversy in the 1990's and 2000's because of the spike in home runs at the major league level, including Mark McGwire's monster season in 1998. Universal Medical Systems tested his 70th home run ball and claimed that a rubber ring around the ball's core enhanced the flight distance (Reference 4). Baseball officials claimed that the baseball manufacturing process had been consistent since 1977 (Reference 4) and, later, McGwire would admit using steroids during that season (Reference 5).