Do gravitational waves exist at the quantum level? Do gravitational waves exist at quantum level produced by electrons or by atoms or by molecules? If yes, which particle produces them and how are they produced?
Spacetime exists at a quantum level and it is curved by the presence of matter (nucleus and electrons), what effect does this curved spacetime have on the movement of electrons or more generally what effect does this curved spacetime have at the quantum level?
 A: General relativity predicts gravity waves.  If subatomic particles deliver the full menu of gravitational effects, they should agree with the equivalence principle, an important corollary of gravity.  In Einstein's own words:

...the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:
(Inertial mass)⋄(Acceleration) = (Intensity of the gravitational field)⋄(Gravitational mass).
It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.

(from the Wikipedia article on Equivalence principle)
Experiments have been performed which seem to confirm that gravitational fields affect sub-atomic particles.  But a theory of quantum gravity will not agree with general relativity unless the equivalence principle also holds true at the quantum level.
A paper written by Mario Rabinowitz, retired researcher at SLAC, calls into question the equivalence principle at the quantum level.  The paper argues that "quantum mechanics...directly violates the strong equivalence principle unless it is arbitrarily abetted in hindsight. Vital domains are shown to exist in which quantum gravity would be non-applicable."  The paper concludes that: "The inevitable ramification of the strong equivalence principle is that gravity is exclusively due to the geometry of space-time curvature, but this appears not to be the case at the quantum level." (https://arxiv.org/abs/physics/0608193)
The Rabinowitz paper refers to neutron interference experiments, both free-falling and in a gravitational field, which it says do not support the equivalence principle.  See section 6 of the paper.
Although the equivalence principle may not hold at the quantum level, gravitation does affect subatomic particles.  One experiment that used ultra cold neutrons found that the phase of the neutron wave function is affected by gravity (Greenberger, D. M.; Overhauser, A. W. The role of gravity in quantum theory; Scientific American, vol. 242, May 1980, p. 66-76).  Another experiment bounced ultra cold neutrons off a hard surface and found they were "sensitive to gravity-like forces" (https://doi.org/10.1016/j.nuclphysa.2009.05.131).  Unfortunately this paper is behind a pay wall.
Here is a link to a paper that analyzes the second experiment and finds evidence of a quantum effect caused by gravity: https://faculty.csbsju.edu/frioux/neutron/neutron.htm.  The paper includes a description with diagrams of the experiment.  Although the equivalence principle is not mentioned, the author cites the experiment as "direct evidence for quantized gravitational states."
Subatomic particles are affected by gravity, but do they themselves gravitate, and are they capable of creating gravity waves?  So far as I know, at the present time there is no conclusive evidence of this.
A: String theories quantize  gravity  and aim to a model of a unified theory with the other three interactions which are well described by the standard model for elementary particles. 

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.

There does not yet exist such a string model, which should contain the standard model and extensions that would explain discrepancies ( like CPviolation and baryon antibaryon asymmetry ) due to the large number of possible models.

One of the challenges of string theory is that the full theory does not yet have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and this has complicated efforts to develop theories of particle physics based on string theory. 

Nevertheless, the graviton will exist in any string  model and will be represented in the equivalent field theory. Thus similar to the way that classical electromagnetic waves emerge from a confluence of photons, and generally classical fields from the quantum mechanical operators, one would expect that gravitational waves would naturally emerge from a confluence of gravitons.
There is strong research effort ongoing in this field.
A: To complement Ernie's description of the experimental evidence (and some of the theoretical evidence) that gravity must ultimately be 'quantum' in nature there are additional theoretical arguments that gravity waves (which as Ernie said are distinctly predicted by general relativity, even though we haven't observed them yet) must correspond to some microscopic quanta (analogously to how electromagnetic waves correspond to photons).
One could suppose that the ultimate resolution of the tension between general relativity and quantum mechanics is that gravity is ultimately classical. This is generally referred to as semiclassical gravity, but semiclassical gravity has at least one severe logical problem (which is mentioned in the wikipedia article) which is that superpositions of large masses would source gravitational potentials as if they were smeared according to the wavefunction of the mass, but measurement would cause a non-local collapse and cause the matter to 'vanish' from one of the two points, which, although not a big deal quantum mechanically, would make the stress-energy tensor sourcing the classical graviy violate local conservation of energy-momentum, which is more than just a pleasant feature of GR but rather necessary for the full diffeomorphism invariance of GR. So if you try to deal with matter with a stress-energy tensor that violates the local conservation of energy-momentum you get a system who's physics depends on what coordinate system you write it in, which is one of the fundamental assumptions that goes into GR. (Although strictly speaking there's a way around the whole issue which is assuming that the many worlds interpretation of QM is true and there just is no physial wavefunction collapse. This would then mean that the parallel universes experience eachother's gravity, which would at least be a testable prediction, although this is far from the only logical consistency problem with semiclassical gravity.) Another problem is that the semiclassical formalism requires a Cauchy surface of initial conditions or in other words a time slicing of the spacetime in question, which is sort of tantamount to constructing a global time variable and somewhat unnatural in GR (and prone to logical problems). A final problem is that the time evolution of matter's wavefunction would be nonlinear, although this is less of a fundamental concern in that it's at least in principle consistent (although if I remember correctly non-linear QM effects would allow you to construct so called 'post-selection' which essentially allows you to choose which state a measured wavefunction collapses to and this allows you do many things that people find suspicious, although again not downright inconsistent). 
The more 'quantum' way to resolve the issue, of course, is to say that the spacetime geometry itself develops a superposition corresponding to the two possible locations of the mass. This means that the metric needs to be described by some kind of quantum operator and quantum waves essentially need to have microscopic quanta associated with them (I say essentially because there are things like conformal field theories which don't properly have particles, but they're not direct descriptions of any fundamental physics, mostly normal quantum systems at critical points like phase transitions). Furthermore there's some reason to believe that the graviton itself cannot be a composite particle, which would rule out more indirect or emergent description of gravity.
