Can photons only exist in the state of motion in a medium? Photons are known to travel at a speed of $\rm 299 \ 792 \ 458 \ m / s$ in vacuum. Photons can be absorbed, or absorbed and re-emitted by matter. They slow down to  $\rm 225,000,000 \ m/s$ in water with a refractive index of $1.3$, to $\rm 200,000,000 \ m/s$ in glass and to $\rm 125,000,000 \ m/s$ in diamond. So its speed can be varied with variations in the medium.
What happens if the photons can be virtually brought to a stop and is not absorbed by matter? Can the photon exist in a state where it has an energy of  $\rm E = h \nu$,  but having no velocity?
 A: You are confusing photons with electromagnetic waves, light.

They slow down to 225,000,000 m/s in water with a refractive idex of 1.3, to 200,000,000 m/s in glass and to 125,000,000 m/s in diamond. So its speed can be varied with variations in the medium.

This is not correct . Light,the electromagnetic wave, slows down. Light is made out by the superposition of a large number of photons, but photons are not light. They are elementary particles with mass zero and will always , when they exists, move with the speed c of light in vacuum, even within matter.
See this simple experiment


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

to understand how photon wave functions interfere and show the interference fringes expected by light, even though the individual footprint is a  dot in (x,y,z,t). The wave nature of photons is a probability wave, it needs an accumulation of photons to appear.
In a medium individual photons will interact  randomly at random angles, always at the speed of light ( more complicated than  hitting the screen in the link)  they will take complicated paths . If the medium is transparent and the classical light does not change color, it means they will make a large number of elastic collisions which  extend the path length traveled by individual photons and will be seen as the slowing of the  electromagnetic wave, light.
A: It depends how you're defining "motion". As you're talking about photons as opposed to light generally, I'll talk from the quantum perspective rather than just classical waves. See these related questions for more on the different perspectives: 
Do photons actually slow down in a medium, or is the speed decrease just apparent? 
What is the mechanism behind the slowdown of light/photons in a transparent medium?
Remember that whilst we say that photons/light slows down in a medium, the individual photons still locally propagate at $c$ in between interactions (described by QED). So slowing photons down means increasing the number of interactions with other forms of matter (i.e. absorption and emission). Getting to your question:

What happens if the photons can be virtually brought to a stop and is
not absorbed by matter?

The answer is that either that they're absorbed and not emitted, which doesn't seem to mean what you intended, or the answer to your title question is yes, the massless photons can only exist in a state of motion. "Brought to a stop" doesn't make sense if we're talking about photons themselves.
A: Perhaps this question is answerable in a purely classical electromagnetic wave setting.
You need to be aware that there are different definitions of "speed" for waves.
The speed of light is a "phase velocity", which is not really a velocity (see below).
The phase velocity is more the ratio between frequency and wavelength. Both are physically well defined and measurable parameters.
The different speed of light (or refractive index) is now, in its essense, just a measure that a wave has different wavelength at same frequency in some media.
So why it is dangerous imagining the phase vecloity "speed of light" as an actual speed?:

*

*An  (infinitely long) light beam does not have a "speed". Still the "speed of light" is well defined and changes with medium.

*A wave-packet e.g. a a light burst (or the front of a just-turned-on light beam), actually does not generally propagate with a speed equal to the phase velocity. The right term is the group velocity or the front velocity of the wave. Just in vaccuum these velocities happen to be the same. But they are generally not. They are changed by the medium and the waveguide the light is confined in.

*In a wave packet the ratio between wavelength and frequency is still the phase velocity, but the envelope may move much slower (or faster) in a medium and/or waveguide.

*Consider a cavity, that is in its simplest form just two parallel mirrors. With hypothetical ideal mirrors one can store light or photons in it with an intensity distribution which is stable over time. So there is no "movement".

*Perhaps you want to check out the Slow Light field where scientists try to design structures which allow for extremely low group velocity. This sounds more fancy than it is. Essentially this works by some sort of resonator or cavity where the light is circulating  or "stored".

So to answer your question: Yes, in a ideal resonator photons can be brought to stop without being absorbed. The energy-frequency relationship for photons is not affected and still valid.
Your confusion most likely is due to the definition(s) of velocity. Velocity is not well defined for a single photon, but only for waves, and even there different velocities exist, accounting for the nature of waves.
Note: In the RF there are superconducting cavities, which come very close to the ideal, lossless resonator.
