The poster from that link is saying that the work done by the spring (that's Hooke's law there: $F=-kx$) is equal to the potential energy (PE) at maximum displacement, $A$; this PE comes from the kinetic energy (KE) and is equal to the integral of Hooke's law over the range 0 (minimum displacement) to $A$ (maximum displacement).
Anyway, your professor is wrong. The total energy in a wave comes from the sum of the the changes in potential energy,
$$\Delta U=\frac12\left(\Delta m\right)\omega^2y^2,\tag{PE}$$
and in kinetic energy,
$$\Delta K=\frac12\left(\Delta m\right)v^2\tag{KE}$$
where $\Delta m$ is the change in mass. If we assume that the density of the wave is uniform, then $\Delta m=\mu\Delta x$ where $\mu$ is the linear density. Thus the total energy is
$$E=\Delta U+\Delta K=\frac12\omega^2y^2\,\mu\Delta x+\frac12v^2\,\mu \Delta x$$
As $y=A\sin\left(kx-\omega t\right)$ and $v=A\omega\cos(kx-\omega t)$, then the energy is proportional to the square of the amplitude:
$$E\propto\omega^2 A^2$$
1. How do I know what type of wave is travelling when the impact happens? Would it be longitudinal or transverse?
For very thin solids usually you will have a mixture of longitudinal and transverse motion occurring. It would be inaccurate to try and classify the motion of a membrane as either case individually.
2. Is the speed of sound in the material same thing as wave speed in the material? Sometimes it is used interchangeably and it is confusing.
Calling mechanical wave motion in solids "sound" is a little confusing, as we typically associate sound with something we hear, which intrinsically is associated with fluids. However, for the most part you can use "wave speed" and "sound speed" interchangeably even for solids (unless they are specifying a non-mechanical wave). It is important to remember that there are two different wave speeds in bulk solids (compressive and shear), and neither of these are the wave speed for thin membranes.
3. Are waves created due to impacts always acoustic waves? I think the waves in the foil might be acoustic since it travels through a medium and I can hear it but not see it.
Similarly with the discussion for "sound", "acoustic" can be used to describe small-amplitude (no nonlinear effects) mechanical vibrations in solids as well as fluids.
4. Are acoustic waves a type of longitudinal wave or is the the other way around ? I could not find a proper classification tree for the wave types.
If someone is trying to make a distinction, acoustic refers to longitudinal (compressional). However, there are plenty of cases when "acoustic" can also refer to transverse (shear) waves.
5. Can I say that if I have an acoustic wave then it is a longitudinal transmission of waves?
See my response to 4.
6. Is it possible that I have a combination of longitudinal and transverse waves when the impacts happen?
Not only is it possible, it is almost certainly what is happening.
7. Do the waves have different speeds of propagation in x and y direction? Is there an equation to determine this?
In the bulk, this only happens if the material is different in one direction than another (think of wood along the grain versus against the grain). In a membrane you get a lot of your wave motion from the tension of the membrane; if you pulled one side tighter than the other, the wave speeds will be different in the two directions.
8. Can I only determine the speed of waves in a material if I have a standing wave? experimentally or theoretically
No, you can readily determine the speed of waves using propagating, non-standing, waves. One approach that might be helpful is to measure a propagating signal at two points and do a cross-correlation between the two signals. The peak in the time difference gives you how long it took the wave to get from the one measurement site to another. However, since you have a bounded domain, my guess is that you will wind up using standing waves if you are doing this experimentally.
Theoretically, if the foil is truly thin enough to be considered a membrane (which it probably is), then you can just use the formula for the wave speed in a taut membrane:
$$c = \sqrt{T/\sigma},$$ where $c$ is the wave speed, $T$ is the tension of your membrane and $\sigma$ is the areal mass density (mass density of your material times the membrane thickness).
9. To calculate the wave speed using the formula of $\sqrt{B/\rho}$, is it necessary that I know the type of wave that is travelling in my medium? That is whether it is longitudinal or transverse?
That expression is a generic expression for waves in a bulk, where $B$ could be the shear modulus for transverse waves or it could be a mixture of shear and bulk moduli for compressional waves. You need to know what you are working with to use it appropriately. But, remember that waves on a membrane are not in the bulk, so don't use this equation for your case at all. See the answer to 8.
10. Is there a difference in formula when calculating wave speed for a solid vs a membrane type of material ?
Yes; see my answer to 8.
Best Answer
The transverse speed is the rate at which the orthogonal components of the wave move.
Treat the derivation of the transverse speed equation as SHM, as you mention in your post.