A sound wave is not particles oscillating, is a mechanical oscillation of a medium made of particles. It is important to separate the medium behaviour from the particle behaviour. Medium behaviour results from the averaging over the values of the many particles, and this generates essentially different phenomena.
A way of visualizing it is thinking of a demonstration or a public gathering: although many people are moving through, trying to reach a friend or coming out or in, the bulk behaviour is what matters if you would see from a plane. From above the mass would look static, even if below there is almost nobody standing in one place. The same is with movement, although not every person might move in the direction of the bulk, from above a marching crowd would look so. But the speeds and directions might differ very much.
So when sound travels through a medium, average densities oscillate due to pressure increase and vice versa. But local densities at a microscopic scale might me much larger than the bulk ones, because even if two particles can come very close, many of them cannot be so close together due to the much higher potential energy related.
So your analogy cannot be followed for these reasons, and this is also the cause that we use different formulations to describe groups of 10 or 100 particles, than when we describe media (made of at least ~$10^{23}$).
Because the frequency of a sound wave is defined as "the number of waves per second."
If you had a sound source emitting, say, 200 waves per second, and your ear (inside a different medium) received only 150 waves per second, the remaining waves 50 waves per second would have to pile up somewhere — presumably, at the interface between the two media.
After, say, a minute of playing the sound, there would already be 60 × 50 = 3,000 delayed waves piled up at the interface, waiting for their turn to enter the new medium. If you stopped the sound at that point, it would still take 20 more seconds for all those piled-up waves to get into the new medium, at 150 waves per second. Thus, your ear, inside the different medium, would continue to hear the sound for 20 more seconds after it had already stopped.
We don't observe sound piling up at the boundaries of different media like that. (It would be kind of convenient if it did, since we could use such an effect for easy sound recording, without having to bother with microphones and record discs / digital storage. But alas, it just doesn't happen.) Thus, it appears that, in the real world, the frequency of sound doesn't change between media.
Besides, imagine that you switched the media around: now the sound source would be emitting 150 waves per second, inside the "low-frequency" medium, and your ear would receive 200 waves per second inside the "high-frequency" medium. Where would the extra 50 waves per second come from? The future? Or would they just magically appear from nowhere?
All that said, there are physical processes that can change the frequency of sound, or at least introduce some new frequencies. For example, there are materials that can interact with a sound wave and change its shape, distorting it so that an originally pure single-frequency sound wave acquires overtones at higher frequencies.
These are not, however, the same kinds of continuous shifts as you'd observe with wavelength, when moving from one medium to another with a different speed of sound. Rather, the overtones introduced this way are generally multiples (or simple fractions) of the original frequency: you can easily obtain overtones at two or three or four times the original frequency, but not at, say, 1.018 times the original frequency. This is because they're not really changing the rate at which the waves cycle, but rather the shape of each individual wave (which can be viewed as converting some of each original wave into new waves with two/three/etc. times the original frequency).
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When pitch of a voice is changed, both the wavelength and frequency change. For example, a higher pitch will have a higher frequency (of course) but a smaller wavelength.
Okay, that being said, the independence of the propagation speed of a wave to the properties of the wave itself it an model to use in many circumstances, but not all. I don't know about acoustics, but one example that comes to my mind is in the field of optics; you might find dispersion a useful search term.
And for a more general advice, be careful when using equations involving more than two variables. Think about what is varying and what isn't; the equation alone won't tell you.