1. Sports balls are interesting beasts with respect to aerodynamics. Baseballs, footballs, golf balls and yes, tennis balls don't have the design they do by accident.
The bernoulli effect with these items is influenced not only by shape but by whether it spins when used. The math is ugy, they didn't teach us the real equations until my college aerodynamics course, but in concept lift is created by the top of the airfoil creating a temporary vacuum, the bottom not and there is a turbulence effect involved that helps create the vacuum that depends on the shape of the airfoil and its behavior.
Tennis balls are round, and they're designed to travel straight through the air for the most part (sometimes you lob them, but mostly the idea is to have them go fast and straight). I don't know enough about tennis to know if you deliberately put spin on it, as you do in table tennis, baseball etc to make the ball move strangely but it seems likely.
With a baseball, you do a curve ball, fastball or knuckleball by varying the spin on the ball as you throw it...the stitching and the exact weight for our atmosphere causes it to create the right kind of turbulence to change the direction of travel. A tennis ball might be similar, with the "fuzz" contributing to the turbulence, to make "spin" more effective.
But I'm guessing here. I think to see a different Bernoulli effect between a hairy ball and a "bald" ball, you'd want to hit the ball in a way that causes "spin". But I've got nothing to back that up except some half remembered theory of how baseballs work.
Find somebody good at tennis and ask what kinds of strokes they use, whether they ever try to put "spin" on the ball and if they've ever noticed that some balls are better than others for this.
2. The "bounce" of the ball is based on its construction, and the surface it hits (clay, grass, concrete, tennis racket) and the force exerted. The "fuzz" isn't going to be significant compared to those things. I'd expect, in absence of "spin" the ball to have a little more drag with fuzz than without, but it might not be measurable with the tools you have.
If you're measuring "bounce" by dropping the ball from a known height, if the fuzz really exerted enough drag to slow the ball before it hits, that would mean less force exerted, and it would bounce up a lower height. You can try it but my guess is it won't be measurable with tools you have.
3. That's a good question. I can think of two ways offhand but I can't predict how well they'll work. One would be to get a mens electric razor (a new one, don't use your dad's) and try to "shave" it off. Those are designed to work on rounded surfaces and would be better and safer than a knife or straight razor blade. But you might ruin the electric razor or it might prove ineffective.
The other way would be to drop the ball in some kind of acid that dissolves the "fuzz" but not the rest. That should not be attempted without doing reseach on the materials of the ball and without adult supervision, as the acid itself, the fumes and handling of the ball are potentially very dangeorus.
Seriously though, I'd talk to somebody who plays tennis and ask them about how the tennis balls are used. It may well be that the only reason for the fuzz is to affect spin behavior and that may not be something you can set up easily with a drop experiment.