So, I think a mallet is good for finding out how wide and long the sweet spot is. You do the trick of holding close to the bat face, hitting gently with the mallet and seeing if you get a fast vibration of the mallet over a wide area.
Second though, the speed of a ball off the bat definitely determines how high the ball bounces, unless you have imparted some strange spin that changes the aerodynamics. Can give the equation if you like, but it's not very helpful.
Finally, yes I agree that an old ball is best for judging the pinginess off the sweet spot. But, interesting question is whether some bats work better with new balls that worn balls. Could be, say, that hard pressed bats are better for openers and soft bats for middle orders. Anyone got any opinions or evidence about that?