If a laser weapon was fired at a spaceship moving away at near the speed of light , would it lack the energy to damage it when it hit ?

Yes. The energy PER SECOND that the spaceship feels is much less, compared to if it was moving at slower speeds. So the spaceship would have more time to get rid of the energy, probably by simply warming up a tiny bit and then radiating the heat away.

Engineer: "Captain, our hull temperature is up 0.01 degrees"

Captain: "Why you bothering me this stuff?"

How fast does the said laser weapon ejaculate it’s laser, if it’s faster than the spaceship it will reach it

IT would eventually hit the ship. But it's energy diminishes with distance. So it depends on how far it goes until it got the ship and how powerful the laser beam is. 

An interesting note: if the ship is accelerating constantly, the laser beam will never reach it, although the ship never reaches the speed of light. 

irrelevant. Lasers move at the speed of light, and if the intended target moves away at the speed of light (which may be technically impossible...) the laser would never hit it (unless the target slows down)

Yes it would lack any appreciable energy density.

Recall that the wavelength of the light would be enormously redshifted. If you were firing a UV laser at a target, then depending on the speed, the wavelength would be shifted into the infrared, or even radiofrequency wavelengths. The pulse itself would also be greatly spread out (a pulse is just a superposition of waves, all of which are redshifted). Hence the target would be able to either sink or radiate the energy as fast as it was supplied, resulting in no damage.

Yes and hope your throat gets better for Christmas 

It would be like F4rting into  a Black Hole

Yes.  In the frame of the spaceship, the laser would be highly redshifted, resulting in a loss of energy.

No, because the speed at which laser hits does not make much difference, it is not like a physical/boxing punch.