To hit the ball harder, add spin and strike the ball higher and closer to the net to create a better angle into the service box, which increases the margin of error. Bending the knees to jump upward when serving is where most of the power comes from.
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It is exactly what you think, they are trying to identify the most warn ball or balls of the set they choose from.
The more warn a ball is the more fuzzy it will become, and of course that will slow the ball down in it's flight. So it's really just for maximum speed in their serve.
Since new balls are called in roughly every set, the differences between balls are fairly minimal (although noticeable to a trained eye) so the act is more related to habit
Players inspect the balls in order to see which has the least wear on it in order to serve with it. The balls fly faster through the air when the material is still quite compacted, but will be slower when they are fluffier. They can gain an advantage on serve when the ball travels faster, just the same as when they serve with new balls.
They don't get much time to choose, but they usually serve with heavy balls due to the speed of service being afflicted by weight.
Because all tosses are not uniform, due to wind and other factors. They have to know exactly where the ball will be in order to strike it properly.
they carry the extra ball so if they fault their first serve they have the ball ready for the second serve.
Because the balls lose some tightness and bounce after several games; not noticeable to most folk but to professionals it supposedly is.