Can Dark Matter be light and cold at the same time? In most of the papers that I encounter, the mass range for Dark matter is generally taken to be 1-1000 GeV. Is it possible that dark matter is of the order of few MeV? What observation is violated in this case? 
To be clear, I emphasize that the DM is cold (kinetic energy << mass) as is required for structure formation.  
 A: The dark matter you are talking about is "cold dark matter". Cold dark matter is non-relativistic (their kinetic energy is much less than their rest mass energy). The classification refers not to their kinetic energies now (which have been reduced by the expansion of the universe), but their kinetic energies at an epoch that was important for structure formation.
The wikipedia page on cold dark matter suggests that this is about 1 year after the big bang when the particle horizon contains the mass of a typical galaxy. At this epoch temperatures were of order 1 keV, so that would seem to admit the possibility of particles with rest mass energies of $\sim 1$ MeV being described as "cold".
Therefore the reason that cold dark matter candidate particles are expected to have masses $>1$ GeV is down to theoretical ideas. For instance the lightest WIMPS are expected to be of order 100 GeV because they are thought to be the leftovers of an electroweak interaction in supersymmetric models that appear at these energy scales.
On the other hand axions are not expected to be heavy, with masses $<1$ eV. These are still "cold dark matter" candidates however, because they are bosons and could form a condensate with low kinetic energy even at relatively high temperatures. Hence this gives an answer to the title of your question - yes it is possible for light particles to be cold dark matter.
In recent years there has been a move towards looking at/for other particles  such as sterile neutrinos, with masses of maybe keV or tens of keV. However, these are known as warm dark matter and would lead to different observational properties from cold dark matter with regard to their dynamical behaviour and role in structure formation.
As far as I know there would currently be no observational objection to dark matter being made of 1 MeV, non-interacting, particles - they would be cold dark matter.
