Did we ever find a meteorite (a meteoroid that has fallen down to Earth) whose mass, density and gravity were high enough to demonstrate their gravity similar to the Cavendish experiment, so that something was attracted to it?
Was there ever a meteorite with measurable gravity?
At risk of stating the obvious: Yes. All mass, as far as every test I know about, has measured gravity. No test has ever measured no gravity within the range of accuracy, so even low mass objects can have their gravity measured.
Rubble pile asteroids couldn't form if they didn't exert gravity. Asteroids have been observed in orbiting pairs, so there's little question that they exert measurable gravity, if you can get close enough and have good enough equiptment to one to measure it.
Did we ever find a meteorite (a meteoroid that has fallen down to Earth) whose mass, density and gravity were high enough to demonstrate their gravity similar to the Cavendish experiment?
Cavendish measured gravity using 158 kg lead balls and some sensitive equiptment. These larger lead balls were stationary sources of perpendicular gravity on two smaller, 0.73 kg lead balls as the suspended objects on a wire upon which the gravity acted. By the twisting of the suspension wire, and from a distance, so his mass wouldn't affect the experiment, Cavendish measured gravity.
A 158 kg lead ball is about 15 cm in radius and the smaller balls about 2.5 cm. The distance between them (and I'm assuming this was center to center as that's standard in physical notations of this type, is listed as about 9 inches or 29.5 cm. That means, surface to surface the distance was about 12 cm, which doesn't change the equation, but you wouldn't want to calculate gravity with two surfaces inside each other, even if the math lets you get away with that, so, I thought I'd mention it.
Some quick math and 29.5 cm, 158 KG, the gravitational force is easy to calculate. F=Gm/r^2, which comes to about 2.7/10,000,000 meters per second squared about 35 million times weaker than earth gravity. A meteor could certainly have that force of gravity pretty easily. While less dense than lead, greater overall could accomplish 1/35 million G forces easily enough.
What does that mean? 1/35 million G forces? Lets call that amount of gravity one Cavendish - for obvious reasons.
Have you ever walked up a flight of stairs, one step at a time? By just taking one step up a flight of stairs, you've reduced the gravitational force acting on you by one Cavendish. A bit more than one, actually. By taking an elevator up one floor works out to about 33 Cavendishes drop in gravity, for every 3 meters. That's how gravity works. Mass increases (roughly speaking) with the cube of the radius. Gravity falls off with the square of the distance, so larger objects not denser objects, usually win in the gravity contest. There are exceptions if there's enough variation in density, obviously. Earth, for example has higher surface gravity than Uranus and Mercury has very nearly equal surface gravity to Mars, but as a general rule, when the overall mass is much different, density matters less. The Sun and the Moon appear about the same size to our eyes and as a result have similar tidal forces, but the Sun exerts hundreds of times more gravity on the Earth than the Moon does.
The weakness of this force indicates how sensitive Cavendish' experiment was for it's time.
How did Cavendish know that there wasn't a mountain range in the distance that would exert equal pull on his test, or maybe some part of the Earth that was heavier on one side of where he lived than the other side?
Lets imagine that there was a mountain 12 km from Cavendish's laboratory, lets pretend this mountain was a 1.5 km high and weighs about 4 trillion kg. That mountain would exert nearly 7 times the gravitational pull on Cavendish's test balls that the his 158 kg stationary lead balls did, so, did Cavendish need to find a flat region of equal gravitational pull in all directions?
Actually, the answer is no, because a mountain in the distance would have generated a gravitational field that would have been part of the initial conditions and he was measuring the change from the initial conditions.
And even if somebody had decided to mess up Cavendish's experiment by putting a mountain 12 miles from his house during his experiment, the effect would have been nearly equal on the two balls so it wouldn't have made much difference. It would just have been a tidal force, and a very small specific force on the torque of the wire that Cavendish was interested in. Cavendish's experiment only works under very specific conditions, one 158 kg weight pulling both suspended balls on the wire in the same clockwise direction so the twist can be measured. That doesn't work with a single large mass. It's the wrong kind of setup.
But to answer your question, forces greater than 1 Cavendish are easy for any relatively large body and a meteor could do that. Bennu, which you mentioned in a comment exerts 6 micro-G, or about 22 Cavendishes. It's radius is about 250 meters, so within about 1100 or 1200 meters, an object that size would exert 1 Cavendish of gravitational force.
Ofcourse, if Bennu was to actually strike the Earth, it would likely decrease, not increase gravity at the spot it hits because it would eject material over a wide area, leaving a crater. Likewise, the effect would be very small, but on the edge of the crater, probably several Cavendish's worth of decreased gravity.
Perpendicular forces like this happen on Earth. As land rises and falls. As glaciers shift, the perpendicular gravitational force near those massive objects changes and this is most noticeable on the ocean. The gravity of the Greenland ice sheet or the Antarctic ice sheet is measurable on sea level and calculating sea level rise if those ice sheets melt (which will take some time), but accurate calculation has to take into account the gravitational pull of the mass of glacial ice on the ocean's sea level.
The mass of a meteorite is generally quite small in the grand scheme of things and it's gravitational force on a nearby blade of grass, for example, while perhaps several Cavendishes, is negligible for all practical purposes, but measurable if you have sensitive enough equiptment. You can now tell your friend when you drive past a mountain range that your car is experiencing a force greater than the force Cavendish first use to measure the gravitational constant and they probably won't know what you're talking about.
And if you're talking a much smaller, mostly Iron meteorite fragment, then you need to be quite close. The largest meteorite fragment or solid piece is the Hoba, about 60,000 KG. You'd need to be about 4 meters from it's center to experience 1 Cavendish of gravitational force.
It's fun to think that walking past a very dense, very heavy object, that you might feel it's gravity, but most things aren't nearly massive enough. You don't feel the gravitational pull of a mountain range when you walk past it because the pull is negligible and mountain ranges are much more massive than most asteroids that are likely to hit the Earth. Mountain ranges also are generally made up of lighter rock than the surrounding area, so there's also that, but even if they weren't, you still couldn't feel the gravitational pull of even the largest string of mountains.
Hope that wasn't too long. I got answering and it came out longer than probably necessary.
That would be very difficult to do. The meteor will pulverize when striking earth and it would be difficult to determine the original size of the meteor. Estimates of the initial mass can maybe be determined from the size of the crater but I doubt the accuracy of this approach. Fragments of the meteorite can be collected and used for the experiments you mentioned, but so could earth rocks!