Will a bound electron absorb energy that is less than a difference between energy levels? Can an electron absorb a photon which has energy less than the difference between any of the energy levels? will it increase its speed without exciting it? or that just doesn't happen. 
And also what happens when an electron absorb a photon that have energy higher than the difference between, let's say 1 and 2, will it be excited and have the excess energy as kinetic energy? or will it be excited and the photon will have the remaining energy?
 A: According to the resonance condition, a photon can only be absorbed by an electron if the energy of the photon corresponds to the energy difference between two energy levels. If the energy of the photon is less, it cannot be absorbed. However, when the intensity of the light is very high, nonlinear effects can induce a two-photon absorption via a virtual level in which case each photon provides half the energy difference.
If the photon's energy is larger than the gap between the levels, the excess energy is only converted into kinetic energy if the photon has enough energy to remove the electron from the system, that is, the photon energy should be higher than the ionzation energy of the system. If not, no transition occurs.
note added:
As a final note, true monochromatic light does not exist and photons from any realistic light source have a certain distribution of energies (even the levels themselves have a distribution due to life time effects for instance), if the width of the distribution is large enough, some photons still have the "right" energy even if the central frequency is off.
