Is there a way to "hold a molecule still"? Is there a way (even if only theoretical) to hold all the nuclei of one single molecule in place, stoping them from vibrating and stoping the molecule completely (0 kinetic energy)? If so, what happens to the electrons when the molecule is trapped like that?
EDIT:
Let me add some more information, in the form of assumptions that I had prior to ask this question:


*

*I assumed all molecules move through space, even in a solid or liquid, even though they move less than a gas molecule.

*I assumed, that there is movement inside the molecules, their nuclei would change their position in relation to an arbitrary point $P_0$ that would be the "origing point of the molecule".

*I assumed it was possible (and very common) to be able to restrict the first kind of movement during physics experiments, but I wasn't sure about the second kind, since, in my naïve assumption, that movement was the result of interactions between the atoms of the molecule.

 A: Surprisingly, one can reduce the vibration to zero by leaving the molecule alone. :-)
I know! That's about as counter-intuitive as it comes. 
The very attempt to trap it will cause the very movement that you want to avoid and as strange as that sounds, we are not dealing with classical objects here but with quantum mechanical ones and the world is, a little bit, upside down here. Black is white, white is black and we have to go down the rabbit hole, like Alice, and learn how life works in Wonderland. This requires an exposure to the actual phenomenology of quantum objects and some slightly non-trivial math (the structural reasons that are behind the uncertainty relation), but I won't talk about that.  
Let's put the most simple case this way: if we had "quantum hands", they would feel very little of vibration if we just held a quantum object lightly. The stronger we tried to grab it, though, the more "violently" it would behave and the more shaking we would feel. If we let go of the object, all sensation would seize, but if we grabbed on too hard, the object would break up into new objects that were not there, before! That is exactly what they are doing at LHC: they are localizing quantum objects "so hard" that new objects come out of the interaction. 
In slightly more technical terms localization in space (the famous "particle in a box") increases the energy/momentum of quantum objects, but that energy is put in by the localizing mechanism, i.e. one has to do work while closing the lid on the box that holds the molecule. The same amount of energy that is needed to localize it would then be released if one lets go of the object, again, i.e. every momentum measurement of a localized object just reminds us that we are the ones who are localizing it, in the first place. That energy didn't come with the object as the phrase "zero point energy" suggests, but it came from us. "We", i.e. our measurement devices that perform the localization are always part of the equation.
So, in essence, if we leave a quantum object alone, and we let it shed all the other forms of energy that is has (e.g. electronic excitations) in form of radiation, it will, eventually, stop "vibrating", at least in the sense of quantum mechanics, where everything that we can learn about an object has to come trough some sort of interaction with it. 
Now, if we are applying what we know about the scale of these effects to macroscopic traps, then it turns out that we have negligible momentum uncertainty unless we go to very low temperatures: $\sigma_x\sigma_p>  \hbar/2$. Let's say we trap a benzene molecule with a molecular mass of $78g/mol$, so that's roughly $78g/{6\times 10^{23}}\approx 1.3\times 10^{-22}kg$ of mass in a $1cm$ sized trap. 
We have then 
$\sigma_x\sigma_p = \sigma_xm\sigma_v \approx 5\times 10^{-35}{m^2kg\over s}$ 
and 
$\sigma_v \approx 5\times 10^{-35}{m^2kg\over s}/(0.01m\times 1.3\times 10^{-22}kg)\approx 3.8\times 10^{-11}m/s$. 
That's not easy to measure considering that at room temperature that molecule will move at hundreds of $m/s$ just due to thermodynamics!
A: A step function that is $1$ on a set of finite measure but $0$ elsewhere is   killed by the momentum operator, and so should be thought of as a state of zero momentum.  And any such state, being killed by the momentum operator $p$, is also killed by the kinetic energy operator $p^2/2$, hence can be said to have kinetic energy zero.  
A: Yes, it is possible to trap a molecular ion.   Paul and Penning traps are well developed for the purpose of holding a single particle (or small bunch) in place.   It is also possible to cool such a molecule, using 'optical molasses', a
laser resonance technique.  The Nobel Prize was awarded in 1989 for work in this field by Norman Ramsey, Wolfgang Paul, Hans Dehmelt. See http://tf.nist.gov/general/pdf/1044.pdf for a description of trapping and cooling.
It has to be an ion, for this to work; you need the electric charge in order to hold the molecule in place.   That means the electron configuration is not precisely the same as a neutral molecule (and it is somewhat perturbed by the local electric field, which holds the ion against gravity).
While the electron configuration isn't the same as a neutral molecule, it IS minimally perturbed by collisions and thermal motion, so some interesting high precision atomic experimentation is possible on such a trapped particle
A: I actually remembered asking this similar question in the University for Software Enginereing.
The answer very briefly is yes, it's very possible. But can it be done conveniently? Not really.
For a molecule to be still, it must be at absolute zero. As heat is often described as the product of released energy via atomic interaction [sic]. So for absolute zero to be touched, you'd need to reduce a molecule's temperature to  –273.15°C or –459.67°F. Which... as said before... not convient.
To do this, so far the only method I have actually heard of was the use of L.A.S.E.Rs and electromagnetism. Which with doctorly precision, can reduce the molecule to stillness.
Since I really bare no weight as a student :P Here's some links explaining the history and method to do so.
http://www.iflscience.com/technology/scientists-freeze-atoms-absolute-zero-microwaves/
https://en.wikipedia.org/wiki/Laser_cooling
