1) While a drum played in a vacuum would not produce sound, it would still vibrate. The frequency of vibration is determined again by the vibrating mass and the stiffness. The lack of air would not appreciably alter the stiffness (at least not if the drum is not a double sided drum) which is provided by the tension in the drum head. However, the motion of the drum head also causes the air in the vicinity of the drum head to move as well, adding the mass of that captive air to the mass of the drumhead. In the vacuum that added mass is not there, and the frequency would rise (slightly).
One of the primary dampling mechanisms for a drumhead is via the radiation of sound. Since no sound would be radiated in a vacuum, that source of damping would vanish, and the total damping would go down.
2) When you fill up the car, you increase the mass of the car, which would cause the oscillation of the car on its springs to decrease in frequency. When the car turned upside down, instead of the whole mass of the car bouncing up and down on the springs, whith the wheels being kept at rest on the road, it would be the wheels which bounce up and down, and the car body would be stationary on the road. Since the wheels have a much smaller mass than the body, the wheels would have a much higher frequency of oscillation than the car.
3) The thicher material of the bell would increase its mass. However the thicker material would also make the bell much stiffer. As with the aluminium bar, the stiffness wins out making the frequency go higher.
4) A tuning fork has tines which vibrate back and forth. The primary parts moving are the ends of the forks ( the tops of the U). Thus decreasing the mass there would be expected to increase the frequency. removing material from the ends would not change the stiffness much (or rather it would make the stiffness go up slightly), and would decrease the moving mass, which would bring the frequency up.
If on the other hand you removed material from the bottom of the U, this is a part of the fork which does not move much during the vibration. This this would not decrease the moving mass. However it would decrease the stiffness (filing away mass from that center area would make the material there thinner and much easier to bend) Thus the mass remains the same, but the stiffness goes down, which also drives the frequency down.
5) In the air vibrating in the bottle, the primary moving mass is that of the air in the neck of the bottle. The stiffness comes about by the compression or rarifaction of the air inside the bottle. Thus emptying the bottle would not alter the moving mass much, but would make the amount of air inside greater. Thus to compress the air inside by a given amount (by the air moving in and out of the neck) would become easier (stiffness goes down). Thus the frequency would go down.
Alteratively, by lengthening the neck, you would increase the moving mass, but not alter the stiffness, and thus the frequency would again go down.
6) If you narrow the neck, the first reaction would be that this decreases the moving mass, which it does. However, this also changes the stiffness. Remember that the stiffness is the amount by which the system pushes back ifyour displace the moving mass by a given amount. With a narrower neck, pushing that air in by a given amount would displace less air inside the bottle that if the neck were wider. But also, the increased pressure inside the bottle would have a smaller area to push on as well. This means that the stiffness actually goes down faster than the mass goes down. As you narrow the neck of the bottle, the frequency of the air vibration goes down. This is called a Helmholtz resonator. H. Helmholtz a century ad a half ago used such tuned hollow brass balls, with small holes in them to form resonators, which he would place in his ear and listen to see whether a sound had the frequency to which which he had tuned that resonator contained in it.
7) If the note vibrates 660 times a second, the time of one vibration (the perios) will be 1/660 of a second, or about .0015 sec. If the period is 8 msec= 8/1000 sec., then the number of times it will vibrate in a second will be 1000/8 times, or 125 Hz.
8) The period is the time over which the sound repeats itself. Thus going for example from teh first minimum at about .0017 sec to the second at .0057 sec, gives a differenceof .004 sec. Thus the period is .004 sec. or 4 ms.(milli-seconds) . The frequency would be 1/.004=250 Hz.
The amplitude can be described in various ways. The peak to peak amplitude is from -1.35 to 1.2 of 2.55. The average amplitude would be closer to about .6