Why is energy in a wave proportional to amplitude squared?

It's not true in general that the energy of a wave is always proportional to the square of its amplitude, but there are good reasons to expect this to be true in most cases, in the limit of small amplitudes. This follows simply from expanding the energy in a Taylor series, $E=a_0+a_1 A+a_2 A^2+\ldots$ We can take the $a_0$ term to be zero, since it would just represent some potential energy already present in the medium when there was no wave excitation. The $a_1$ term has to vanish, because otherwise it would dominate the sum for sufficiently small values of $A$, and you could then have waves with negative energy for an appropriately chosen sign of $A$. That means that the first nonvanishing term should be $A^2$. Since we don't expect the energy of the wave to depend on phase, we expect that only the even terms should occur, $E=a_2A^2+a_4A^4+\ldots$ So it's only in the limit of small amplitudes that we expect $E\propto A^2$.

The other issue to consider is that we had to assume that $E$ was a sufficiently smooth function of $A$ to allow it to be calculated using a Taylor series. This doesn't have to be true in general. As an easy example involving an oscillating particle, rather than a wave, consider a pointlike particle in a gravitational field, bouncing up and down elastically on an inflexible floor. If we define the amplitude as the height of the bounce, then we have $E \propto |A|$. But a realistic ball deforms, so the small-amplitude limit consists of the ball vibrating while remaining in contact with the floor, and we regain $E\propto A^2$.

You could also make up examples where $a_2$ vanishes and the first nonvanishing coefficient is $a_4$.

It's just a sine wave. If frequency is constant, then velocity at the zero crossing is proportional to amplitude, and energy is proportional to velocity squared.

Added, responding to comment:

You need a simpler model, like a 1-dimension mass on a spring (or a small-angle pendulum). Its position x is a sine wave of a certain frequency (you can do the math to get the frequency). Its maximum x in one direction is its amplitude a. Its velocity v is dx/dt, which is 90 degrees out of phase with x. At the center of its swing, x = 0, and v = max. Then clearly if you double a, it will have twice as far to swing in the same time, so v will be doubled. I'm sure you got that.

Now your question is, why is energy $E$ equal $mv^2/2$, i.e. proportional to $v^2$? Well, that's a basic equation, but let me see if I can answer it anyway.

If you drop a weight w from a height h, it has initial potential energy wh, which is transformed into kinetic energy as it reaches the floor at velocity v. If it falls under the constant force of gravity, the distance it falls in a given time t is $gt^2/2$ (time-integral of velocity), and the velocity after that time is $v = gt$. So, if you want to double the velocity it has at the floor, you have to double $t$, right? And if you double $t$, you're going to quadruple the height. That quadruples the energy. I hope that answers the question.

Just thought of another explanation. If you have a spring whose force $f$ is $kx$ where $x$ is the displacement of the end of the spring, and $k$ is its stiffness. Since energy (work) is the integral of $fdx$, the energy $E$ stored in the spring, as a function of $x$ is $kx^2/2$. So there's your energy-amplitude relation.

You are correct, the equation is generalizable to higher dimensions. The equation you gave is simply the sum of the various forms of energy. In the equation

$$dE = \frac{\mu}{2} dx \left( \frac{\partial y}{\partial t}\right)^2 + \frac{T}{2} dx \left(\frac{\partial y}{\partial x} \right)^2 $$

The first term on the right hand side is the kinetic energy and the second term is the elastic potential energy. As Mike said, the kinetic energy is just $p^2 / 2m$. The elastic potential energy at any location on the string is given by adding up all of the force it took to get that piece there (given by Hooke's law); that is

$$F_s (x) = -T y(x)$$ $$U = \int_0 ^y Ty(x) dy = \frac{T}{2} y(x)^2$$

The total energy is then just the sum of the kinetic and potential energies with appropriate modification using the mass density times a infinitesimal length as the mass.

When you move to higher dimensions, you have to account for that in the kinetic and potential energies. The kinetic energy of a portion of the surface is the square of the momentum of that portion ($mv$) divided by the mass of that portion. The potential energy is the spring constant of the membrane divided by 2, multiplied by the square of the displacement from zero.