What is the highest possible frequency for an EM wave? What is the highest possible frequency, shortest wavelength, for an electromagnetic wave in free space, and what limits it? Is the answer different for EM waves in other materials or circumstances? How could waves of this frequency be generated and transmitted, again if that is possible?
 A: It's theorized that the Planck length is the smallest meaningful unit of distance. A wave with that wavelength would have a frequency of $\approx 6.2\cdot 10^{34}\,\text{Hz}$. A gamma ray typically has a frequency of $>10^{19}\,\text{Hz}$. Since the energy of a photon is directly proportional to its frequency, this theoretical upper bound would require vastly more energetic processes than those we presently conceive of. The individual photons involved would each be carrying $41\,\text{joules}$, or $2.56\cdot 10^{20}\,\text{eV}$, of energy.
That's a lot of volts!
A: The highest measured frequencies of EM waves are Gamma-rays and are typically produced from the decay of atomic nuclei.  The most powerful sources of gamma-rays (and usually the sources with the shortest wavelength) are caused by astronomical events.  Recently there was a very strong gamma-ray burst from Cygnus-A, .  It is estimated that the gamma ray burst was the result of the black hole gobbling up something with three times the mass of the Earth.
There is no theoretical upper limit for the frequency of gamma-rays.  To make one bigger than what we've seen so far will require starting with a super-massive black hole and something much larger than the Earth.  Not quite reproducible in the laboratory.
A: String theory assumes that lorentz covariance is a perfect symmetry of our world. If that is true, it means a single photon is allowed to have an arbitrary energy, even greater than Planck length.
You need at least two photons that are not parallel to have a rest frame where something like a Planckian black hole might be generated that will absorb them. But single-photon states cannot be bounded in energy like this in a pure vacuum.
If the vacuum is not pure, presumably the ultra-planckian photon will react with background photons creating black holes in the rest frame and being absorbed by it.
A: 
At what frequency does the wavelength = the height of the wave amplitude ?

At 7.7646 x 1020 Hz. The associated wavelength and amplitude is 3.861 x 10−13 m. The electron Compton wavelength is 2.426 × 10−12 m, which is 2π times this amplitude. 

The wave height = the physical space the waveheight occupy in space, similar to the wavelength. My assumption is that the wave height of the basic electromagnetic wave is fixed.

Correct. Take a look at some pictures of the electromagnetic spectrum, note that the amplitude is the same regardless of frequency, and note that the dimensionality of action h can be expressed as momentum x distance. Also note that the reduced Planck's constant ħ is h divided by 2π. 

It's important to note that an electromagnetic wave is an electromagnetic wave. Some people will tell you it's an electric wave and a magnetic wave which generate one another, and therefore no medium is required. That's not true. Check out the Wikipedia electromagnetic wave article and note this: 
"Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field".
What people call the electric wave is actually the spatial derivative of the electromagnetic wave, while the magnetic wave is actually the time derivative. They are merely two aspects of the same wave, not two different waves. And as Maxwell said, "light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena". When an ocean wave travels through the sea, the sea waves. When a seismic wave travels through the ground, the ground waves. When a gravitational wave travels through space, space waves. See LIGO:
"Albert Einstein predicted the existence of gravitational waves in 1916 in his general theory of relativity. Einstein's mathematics showed that massive accelerating objects (such as neutron stars or black holes orbiting each other) would disrupt space-time in such a way that 'waves' of distorted space would radiate from the source (like the movement of waves away from a stone thrown into a pond)".
The same is true for an electromagnetic wave. When an electromagnetic wave travels through space, space waves.
from: John Duffield
